#7623276

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,29)(14,27)(15,19)(16,24)(18,22)(21,33)(23,34)(25,40)(26,36)(28,32)(30,37)(31,39)(35,38);(2,5)(6,8)(9,10)(11,21)(12,27)(13,18)(14,17)(15,19)(16,26)(20,33)(22,29)(23,31)(24,36)(25,35)(28,37)(30,32)(34,39)(38,40);(9,11,22,38,14,25,30,33,15,26,28,39,12,23,13,24)(10,16,29,34,17,31,32,36,19,21,37,40,27,35,18,20);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,27)(13,18)(14,17)(15,19)(20,24)(21,26)(22,29)(23,35)(25,31)(28,37)(30,32)(33,36)(34,38)(39,40);(9,12,15,14)(10,17,19,27)(11,23,26,25)(13,28,30,22)(16,31,21,35)(18,29,32,37)(20,34,36,40)(24,39,33,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322758
maximal normal subgroup of	#72322758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530472);