#7624847

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1301 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)(10,16)(11,20)(12,15)(14,21)(17,30)(19,34)(22,35)(23,37)(24,36)(26,27)(28,33)(29,31)(32,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,17,22,28)(12,25,15,18)(14,19,27,29)(20,33,35,30)(21,31,26,34)(23,38,24,32)(36,39,37,40);(9,11,21,37,15,24,27,35,13,22,26,36,12,23,14,20)(10,17,31,40,18,32,29,30,16,28,34,39,25,38,19,33);(3,8)(5,7)(10,16)(11,22)(18,25)(19,29)(20,35)(23,24)(31,34)(36,37);(2,6)(5,7)(9,12,13,15)(10,18,16,25)(11,23,22,24)(14,26,27,21)(17,32,28,38)(19,31,29,34)(20,36,35,37)(30,39,33,40)

Polycyclic presentation:

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

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, 7532043);