#7623389

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
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

1655 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)(11,19)(13,25)(14,24)(16,30)(17,33)(20,31)(21,37)(22,38)(23,39)(26,27)(28,35)(32,34)(36,40);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,15)(13,17)(14,18)(16,21)(19,30)(22,36)(23,28)(24,29)(25,35)(26,40)(27,32)(31,37)(33,39)(34,38);(3,8)(5,7)(9,11)(10,16)(12,21)(13,26)(14,27)(15,20)(17,34)(18,36)(19,25)(22,24)(23,38)(28,40)(29,32)(30,33)(31,35)(37,39);(2,6)(5,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,31)(20,21)(22,32)(23,28)(24,29)(25,35)(26,34)(27,36)(30,37)(33,39)(38,40);(10,15)(16,20)(17,28)(18,29)(30,31)(32,36)(33,35)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530585);