#6802566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1747 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27,23,28,21,26)(19,24,30,38,31,39,35,41)(20,32,33,40,34,42,36,25)(29,43,45,48,46,47,44,37);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,37)(26,41)(27,38)(28,39)(32,43)(33,45)(34,46)(36,44)(40,48)(42,47);(11,16)(13,15)(17,20,23,34)(18,25,28,40)(19,29,31,46)(21,33,22,36)(24,37,39,48)(26,32,27,42)(30,44,35,45)(38,47,41,43);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,46,45)(37,47,48,43);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=fh,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, 6709762);