#7625173

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

1250 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)(19,28)(20,24)(21,35)(22,27)(23,32)(25,29)(26,34)(30,42)(31,43)(33,48)(36,40)(37,46)(38,47)(39,44)(41,45);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,18)(19,23)(20,26)(21,25)(22,27)(24,34)(28,32)(29,35)(30,37)(31,39)(33,38)(36,41)(40,45)(42,46)(43,44)(47,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,29,44,20,33,36,46,22,30,45,47,34,31,21,32)(18,23,35,43,26,38,41,42,27,37,40,48,24,39,25,28);(3,12)(7,10)(11,13)(15,16)(19,30)(23,37)(28,42)(31,33)(32,46)(38,39)(43,48)(44,47);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,20,22,34)(18,24,27,26)(19,31,30,33)(21,29,36,45)(23,38,37,39)(25,40,41,35)(28,43,42,48)(32,47,46,44)

Polycyclic presentation:

2^9|c^2=fgh,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,d^c=di,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, 7532369);