#6757241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1798 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,35,22,27,21,26)(19,24,36,46,29,38,30,39)(20,31,32,25,34,40,33,41)(28,42,44,37,43,48,45,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,36)(25,37)(26,39)(27,38)(31,42)(32,44)(33,45)(34,43)(35,46)(40,48)(41,47);(9,11,12,15)(10,14,13,16)(17,20,22,34)(18,25,27,41)(19,28,29,43)(21,32,23,33)(24,37,38,47)(26,40,35,31)(30,44,36,45)(39,48,46,42);(2,6)(5,7)(10,13)(14,16)(19,29)(24,38)(28,43)(30,36)(37,47)(39,46)(42,48)(44,45);(3,8)(5,7)(17,21,22,23)(18,26,27,35)(19,30,29,36)(20,32,34,33)(24,39,38,46)(25,40,41,31)(28,44,43,45)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

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, 6664437);