#7623280

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

1400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=gh,d^2=f,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^c=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, 7530476);