#7624853

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

1303 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)(14,22)(17,30)(19,34)(21,33)(23,35)(24,37)(25,36)(27,28)(29,31)(32,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,21,23,17)(12,26,15,18)(14,19,28,29)(20,30,35,33)(22,31,27,34)(24,32,25,38)(36,40,37,39);(9,11,22,37,15,25,28,35,13,23,27,36,12,24,14,20)(10,17,31,40,18,32,29,30,16,21,34,39,26,38,19,33);(3,8)(5,7)(10,16)(11,23)(18,26)(19,29)(20,35)(24,25)(31,34)(36,37);(2,6)(5,7)(9,12,13,15)(10,18,16,26)(11,24,23,25)(14,27,28,22)(17,32,21,38)(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,c^b=ci,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, 7532049);