#7625119

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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

1241 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,30)(20,34)(21,26)(22,27)(23,42)(24,31)(25,38)(28,40)(29,39)(32,47)(33,43)(35,45)(36,46)(37,48)(41,44);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,28)(21,33)(22,31)(23,32)(25,39)(26,38)(27,30)(29,43)(34,35)(36,42)(37,45)(40,48)(41,46)(44,47);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,35,30,21,36,23,38,22,37,40,24,39,47,41,43)(18,25,44,46,29,19,28,34,27,33,42,32,26,31,45,48);(3,12)(7,10)(11,13)(15,16)(17,21,22,39)(18,26,27,29)(19,25,31,33)(20,36,37,47)(23,40,41,35)(24,43,30,38)(28,44,45,42)(32,34,46,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,27)(19,31)(25,33)(26,29)(28,45)(32,46)(34,48)(42,44)

Polycyclic presentation:

2^9|c^2=fgh,d^2=i,e^2=fi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532315);