#6820868

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=gh,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6728064);