#6822308

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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)(10,14)(11,12)(17,18)(19,30)(20,25)(21,29)(22,27)(23,28)(24,33)(26,37)(31,45)(32,39)(34,42)(35,44)(36,43)(38,48)(40,47)(41,46);(9,10)(11,15)(12,16)(13,14)(17,19,23,33)(18,24,28,30)(20,34,36,47)(21,32,37,22)(25,40,43,42)(26,39,29,27)(31,46,48,35)(38,41,45,44);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,35,23,36,37,46)(18,25,29,44,28,43,26,41)(19,31,22,34,33,48,32,47)(24,38,39,40,30,45,27,42);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,23,37)(18,26,28,29)(19,32,33,22)(20,35,36,46)(24,39,30,27)(25,41,43,44)(31,47,48,34)(38,40,45,42);(10,14)(15,16)(17,22,37,19,23,32,21,33)(18,27,26,30,28,39,29,24)(20,34,46,31,36,47,35,48)(25,42,41,45,43,40,44,38)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=gi,e^2=hi,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, 6729504);