#6874402

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,37,22,27,21,26)(19,28,31,40,30,24,32,39)(20,33,34,25,35,41,36,42)(29,44,47,43,45,48,46,38);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,32)(22,30)(23,31)(25,38)(26,40)(27,28)(33,48)(34,47)(35,45)(36,46)(37,39)(41,44)(42,43);(11,16)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,47)(33,37)(39,48)(40,44);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(19,30)(24,28)(29,45)(31,32)(38,43)(39,40)(44,48)(46,47);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,34,35,36)(24,39,28,40)(25,41,42,33)(29,46,45,47)(38,48,43,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6781598);