#7625068

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624876
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 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,d^c=di,e^c=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, 7532264);