#7624357

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,28)(19,26,32,24)(20,30,23,27)(22,37,36,40)(25,43,44,33)(29,35,47,39)(31,41,34,45)(38,46,48,42);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,34)(21,32)(22,33)(23,31)(25,40)(26,28)(27,45)(29,42)(30,41)(35,48)(36,43)(37,44)(38,39)(46,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,25)(20,31)(21,32)(22,38)(23,34)(24,40)(26,37)(27,42)(28,44)(29,45)(30,46)(33,39)(35,43)(36,48)(41,47);(3,11)(6,9)(10,16)(13,15)(17,19)(18,26)(20,34)(21,32)(22,33)(23,31)(24,28)(25,37)(27,41)(29,46)(30,45)(35,48)(36,43)(38,39)(40,44)(42,47);(2,8)(6,9)(7,14)(13,15)(17,20,21,23)(18,27,28,30)(19,31,32,34)(22,35,36,39)(24,41,26,45)(25,42,44,46)(29,40,47,37)(33,38,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531553);