#7624648

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1321 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)(11,14)(12,23)(13,20)(15,29)(16,18)(17,26)(19,31)(21,37)(22,27)(24,39)(25,38)(28,34)(30,36)(32,35)(33,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,21,25,17)(13,20,16,18)(15,22,28,19)(23,26,38,37)(24,36,32,33)(27,29,31,34)(30,39,40,35);(9,12)(10,17)(11,21)(13,24)(14,25)(15,30)(16,32)(18,33)(19,35)(20,36)(22,39)(23,29)(26,31)(27,37)(28,40)(34,38);(2,5)(6,8)(9,13,14,16)(10,18,11,20)(12,24,25,32)(15,27,28,31)(17,33,21,36)(19,34,22,29)(23,35,38,39)(26,30,37,40);(1,3,4,7)(2,6,5,8)(10,11)(17,21)(18,20)(19,22)(23,38)(29,34)(33,36)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531844);