#6826516

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

1258 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)(17,18,22,26)(19,29,31,24)(20,25,32,38)(21,27,35,40)(23,28,33,41)(30,44,45,37)(34,39,46,42)(36,43,47,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,33,32,23)(21,34,35,46)(25,28,38,41)(27,42,40,39)(30,36,45,47)(37,43,44,48);(2,5)(6,8)(11,16)(13,15)(17,20,23,19,22,32,33,31)(18,25,28,29,26,38,41,24)(21,34,36,30,35,46,47,45)(27,39,43,44,40,42,48,37);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(21,35)(24,29)(25,38)(28,41)(30,45)(34,46)(36,47);(17,21,23,36,22,35,33,47)(18,27,28,43,26,40,41,48)(19,30,32,46,31,45,20,34)(24,37,25,39,29,44,38,42)

Polycyclic presentation:

2^9|a^2=i,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^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, 6733712);