#7624754

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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1212 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

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, 7531950);