#7625015

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

1398 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 7532211);