#6840463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(4,8)(5,6)(9,10,13,15)(11,16,12,14)(17,18)(19,29)(20,33)(21,39)(22,26)(23,28)(24,31)(25,36)(27,32)(30,40)(34,41)(35,42)(37,43)(38,48)(44,46)(45,47);(9,10)(11,14)(12,16)(13,15)(17,19,22,31)(18,24,26,29)(20,34,36,37)(21,35,32,23)(25,41,33,43)(27,42,39,28)(30,46,47,38)(40,48,45,44);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,35,46,22,36,23,38)(18,25,42,48,26,33,28,44)(19,30,21,37,31,47,32,34)(24,40,27,43,29,45,39,41);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(18,26)(20,36)(24,29)(27,39)(28,42)(30,47)(34,37)(38,46);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,35,31,22,32,23,19)(18,27,42,29,26,39,28,24)(20,37,46,47,36,34,38,30)(25,43,48,45,33,41,44,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fh,d^2=gi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6747659);