#6826812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(9,10,13,15)(11,16,12,14)(17,18,22,26)(19,29,31,24)(20,25,32,39)(21,27,36,41)(23,28,33,42)(30,45,46,38)(34,40,35,43)(37,44,47,48);(9,10)(11,14)(12,16)(13,15)(17,19,22,31)(18,24,26,29)(20,33,32,23)(21,34,36,35)(25,28,39,42)(27,43,41,40)(30,37,46,47)(38,44,45,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,23,19,22,32,33,31)(18,25,28,29,26,39,42,24)(21,35,37,46,36,34,47,30)(27,43,44,38,41,40,48,45);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(18,26)(21,36)(24,29)(25,39)(28,42)(30,46)(34,35)(37,47);(9,12,13,11)(10,14,15,16)(17,21,23,37,22,36,33,47)(18,27,28,44,26,41,42,48)(19,30,32,35,31,46,20,34)(24,38,25,40,29,45,39,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=g,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6734008);