#6804326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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

1298 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)(20,31,34,43)(21,26,22,37)(25,36,44,33)(28,45,32,48)(29,39,35,41)(38,46,42,47);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,23,30)(18,24,27,40)(20,32,34,28)(21,35,22,29)(25,42,44,38)(26,41,37,39)(31,48,43,45)(33,46,36,47);(10,13)(14,15)(17,20,21,36,23,34,22,33)(18,25,37,43,27,44,26,31)(19,28,35,46,30,32,29,47)(24,38,39,48,40,42,41,45);(2,6)(5,7)(17,21,23,22)(18,26,27,37)(19,29,30,35)(20,33,34,36)(24,39,40,41)(25,43,44,31)(28,46,32,47)(38,45,42,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(19,30)(20,34)(24,40)(25,44)(29,35)(31,43)(33,36)(39,41)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6711522);