#7624869

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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1299 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)(11,23)(12,27)(13,18)(14,22)(15,20)(16,30)(17,19)(21,29)(24,37)(25,35)(26,40)(28,38)(31,34)(32,36)(33,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,18)(12,28)(13,15)(14,32)(16,26)(17,25)(19,36)(20,23)(21,24)(22,35)(27,37)(29,38)(30,39)(31,40)(33,34);(9,12,29,20,17,31,34,19,15,21,27,10,14,30,16,22)(11,24,38,18,25,39,33,35,13,28,37,23,32,40,26,36);(3,8)(5,7)(9,13,15,11)(10,18,20,23)(12,28,21,24)(14,25,17,32)(16,33,34,26)(19,36,22,35)(27,38,29,37)(30,39,31,40);(2,6)(5,7)(9,14,15,17)(10,19,20,22)(11,25,13,32)(12,30,21,31)(16,27,34,29)(18,35,23,36)(24,39,28,40)(26,38,33,37)

Polycyclic presentation:

2^9|c^2=gh,d^2=i,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,e^d=ei

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, 7532065);