#7623380

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

1676 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)(3,7)(4,6)(5,8)(10,15)(12,22)(13,24)(16,27)(17,28)(18,31)(20,34)(21,33)(23,26)(25,37)(29,38)(30,39)(32,40)(35,36);(1,3,4,8)(2,5,6,7)(10,16)(15,27)(17,29)(18,26)(23,31)(28,38)(30,36)(35,39);(3,8)(5,7)(9,10)(11,17)(12,23)(13,26)(14,16)(15,22)(18,24)(19,29)(20,35)(21,36)(25,31)(27,37)(28,34)(30,33)(32,39)(38,40);(2,6)(5,7)(9,11)(10,17)(12,20)(13,21)(14,19)(15,28)(16,29)(18,30)(22,34)(23,35)(24,33)(25,32)(26,36)(27,38)(31,39)(37,40);(11,19)(17,29)(20,32)(21,33)(28,38)(30,36)(34,40)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530576);