#6804142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,30,40)(20,31,35,47)(21,26,22,37)(25,33,43,34)(28,44,32,46)(29,39,36,41)(38,48,42,45);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19)(18,24)(20,32)(21,36)(22,29)(23,30)(25,42)(26,41)(27,40)(28,35)(31,46)(33,45)(34,48)(37,39)(38,43)(44,47);(10,13)(14,15)(17,20,21,33,23,35,22,34)(18,25,37,47,27,43,26,31)(19,28,36,48,30,32,29,45)(24,38,39,46,40,42,41,44);(2,6)(5,7)(17,21,23,22)(18,26,27,37)(19,29,30,36)(20,33,35,34)(24,39,40,41)(25,31,43,47)(28,45,32,48)(38,46,42,44);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(19,30)(24,40)(28,32)(29,36)(38,42)(39,41)(44,46)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6711338);