#7623021

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,21,23,47)(12,18,26,37)(14,31,29,48)(15,34,30,20)(17,39,36,24)(19,44,38,33)(22,49,50,69)(25,53,35,45)(27,56,54,70)(28,59,55,43)(32,46,61,41)(40,64,62,51)(42,66,63,58)(52,71,60,67)(57,68,72,65);(1,3,4,8)(2,5,6,7)(10,16)(17,36)(18,37)(19,38)(20,34)(21,47)(31,48)(40,62)(41,46)(42,63)(43,59)(45,53)(49,69)(56,70)(65,68)(67,71);(9,11,13,23)(10,17,16,36)(12,22,26,50)(14,32,29,61)(15,35,30,25)(18,40,37,62)(19,45,38,53)(20,46,34,41)(21,48,47,31)(24,44,39,33)(27,57,54,72)(28,60,55,52)(42,67,63,71)(43,68,59,65)(49,70,69,56)(51,66,64,58);(9,12)(10,18)(11,22)(13,26)(14,27)(15,28)(16,37)(17,40)(19,42)(20,43)(21,49)(23,50)(24,51)(25,52)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,62)(38,63)(39,64)(41,65)(44,66)(45,67)(46,68)(47,69)(48,70)(53,71)(61,72);(12,26)(18,37)(22,50)(27,54)(28,55)(40,62)(42,63)(43,59)(49,69)(51,64)(52,60)(56,70)(57,72)(58,66)(65,68)(67,71)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530217);