#7625159

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=i,e^2=f,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,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532355);