#7624905

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1310 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)(9,10)(11,14)(12,23)(13,18)(15,31)(16,21)(17,27)(19,24)(20,33)(22,29)(25,37)(26,36)(28,40)(30,34)(32,39)(35,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,20,26,17)(13,18,16,21)(15,22,30,19)(23,27,36,33)(24,34,29,31)(25,32,28,35)(37,38,40,39);(9,12,24,38,16,28,30,33,14,26,29,39,13,25,15,27)(10,17,31,37,18,32,22,36,11,20,34,40,21,35,19,23);(3,8)(5,7)(9,13,14,16)(10,18,11,21)(12,25,26,28)(15,29,30,24)(17,32,20,35)(19,31,22,34)(23,37,36,40)(27,39,33,38);(2,6)(5,7)(10,11)(12,26)(18,21)(19,22)(25,28)(27,33)(31,34)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^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,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, 7532101);