#7624777

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624170
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

1207 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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=dfi,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531973);