#6643291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=fh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efh,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6550487);