#7624353

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,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, 7531549);