#72249772

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#23573
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#23573
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,13)(4,18,23,9)(5,12)(6,10)(7,11)(8,14)(15,33,43,28)(16,41)(17,42)(19,31,54,53)(20,36,24,32)(21,37)(22,39,27,35)(25,38)(26,40)(29,44)(30,45)(34,55)(46,56,64,63)(47,60,50,57)(48,58)(49,62,51,59)(52,61);(3,14)(4,19,23,54)(6,21)(7,26)(8,13)(9,31,18,53)(10,37)(11,40)(15,46,43,64)(16,48)(20,22,24,27)(25,55)(28,56,33,63)(29,61)(32,39,36,35)(34,38)(41,58)(44,52)(47,49,50,51)(57,62,60,59);(4,20,23,24)(6,25)(9,32,18,36)(10,38)(15,47,43,50)(16,44)(19,22,54,27)(21,55)(28,57,33,60)(29,41)(31,39,53,35)(34,37)(46,49,64,51)(48,52)(56,62,63,59)(58,61);(1,3,5,14)(2,8,12,13)(4,15)(6,16,25,44)(7,17,26,45)(9,33)(10,29,38,41)(11,30,40,42)(18,28)(19,46)(20,47)(21,52,55,48)(22,49)(23,43)(24,50)(27,51)(31,63)(32,60)(34,61,37,58)(35,62)(36,57)(39,59)(53,56)(54,64);(1,4,21,19,26,20,6,27,5,23,55,54,7,24,25,22)(2,9,34,31,40,32,38,35,12,18,37,53,11,36,10,39)(3,15,48,64,45,47,44,49,14,43,52,46,17,50,16,51)(8,28,58,63,42,57,29,62,13,33,61,56,30,60,41,59)

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=h,d^2=hj,e^2=fgij,g^2=i,i^2=j,b^a=bh,d^a=dj,d^b=dhj,e^a=ef,e^b=efgij,e^c=eij,e^d=efj,f^a=fj,f^d=fj,f^e=fj,g^b=gi,g^c=gj,g^e=gj,h^e=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249843, #72307129, #72307305, …
maximal normal subgroup of	#72249843, #72307129, #72307305, …
maximal quotient of	#72250221, C₂×#72249772

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);