#7623969

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

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

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, 7531165);