#7624750

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
prime radical	#62495	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1199 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)(17,18,20,24)(19,27,28,46)(21,33,31,45)(22,36,32,26)(23,38,39,29)(25,42,41,35)(30,47,37,43)(34,44,48,40);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,39)(30,40)(31,41)(32,36)(33,35)(34,43)(37,44)(38,46)(42,45)(47,48);(17,19,20,28)(18,23,24,39)(21,34,31,48)(22,37,32,30)(25,43,41,47)(26,44,36,40)(27,45,46,33)(29,42,38,35);(3,10)(6,9)(11,16)(14,15)(18,24)(23,39)(25,41)(26,36)(27,46)(33,45)(40,44)(43,47);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,24)(23,39)(25,41)(26,36)(27,46)(33,45)(40,44)(43,47)

Polycyclic presentation:

2^9|a^2=f,c^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, 7531946);