#72307295

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#24089
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#368058
prime radical	C₂×#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=e,c^2=fgj,d^2=ij,g^2=ij,i^2=j,b^a=bf,c^a=ch,c^b=cg,d^a=dfhj,d^b=dfhij,d^c=dfh,e^d=ej,f^c=fj,f^d=fj,g^b=gi,g^c=gj,h^b=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72343942, #72343980
maximal normal subgroup of	#72307301, #72343942, #72343980

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