#6836404

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1392 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=fi,e^2=ghi,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, 6743600);