#72249778

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21408
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#21408
prime radical	#10575319	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5073 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)(3,9)(4,18)(5,24)(6,11)(7,12)(8,13)(10,23)(14,29)(15,36)(16,32)(17,33)(19,34)(20,46)(21,30)(22,45)(25,51)(26,31)(27,38)(28,37)(35,48)(39,52)(40,53)(41,60)(42,55)(43,57)(44,56)(47,58)(49,59)(50,54)(61,63)(62,64);(3,14)(4,19,23,22)(5,25)(6,28)(7,27)(9,29)(10,34,18,45)(11,37)(12,38)(15,40,21,42)(16,43)(20,47,48,49)(24,51)(26,44)(30,53,36,55)(31,56)(32,57)(35,58,46,59)(41,61,50,62)(54,63,60,64);(4,20,23,48)(5,6)(10,35,18,46)(11,24)(15,41,21,50)(16,26)(19,47,22,49)(25,28)(30,54,36,60)(31,32)(34,58,45,59)(37,51)(40,61,42,62)(43,44)(53,63,55,64)(56,57);(1,3)(2,9)(4,21)(5,26)(6,16)(7,17)(8,14)(10,36)(11,32)(12,33)(13,29)(15,23)(18,30)(19,42)(20,50)(22,40)(24,31)(25,43)(27,39)(28,44)(34,55)(35,60)(37,56)(38,52)(41,48)(45,53)(46,54)(47,62)(49,61)(51,57)(58,64)(59,63);(1,4,6,22,7,20,28,47,8,23,5,19,27,48,25,49)(2,10,11,34,12,35,37,59,13,18,24,45,38,46,51,58)(3,15,16,42,17,41,44,61,14,21,26,40,39,50,43,62)(9,30,32,53,33,54,56,64,29,36,31,55,52,60,57,63)

Polycyclic presentation:

2^10|b^2=h,c^2=h,e^2=g,f^2=i,g^2=i,i^2=j,b^a=bh,d^b=dhj,e^a=ej,e^b=ef,e^c=ei,e^d=ej,f^a=fj,f^b=fi,f^c=fj,f^d=fj,g^b=gi,g^c=gj,h^e=hj,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249843, #72307129, #72307207, …
maximal normal subgroup of	#72249843, #72307129, #72307207, …
maximal quotient of	#72257024, C₂×#72249778, #72323088

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, 2, F.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.9);;
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, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, 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, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);