#6831855

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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)(17,18,23,28)(19,24,31,39)(20,25,34,42)(21,26,35,43)(22,27,30,38)(29,37,46,48)(32,40,36,44)(33,41,45,47);(9,10)(11,14)(12,16)(13,15)(17,19,23,31)(18,24,28,39)(20,32,34,36)(21,22,35,30)(25,40,42,44)(26,27,43,38)(29,45,46,33)(37,47,48,41);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,21,33,23,34,35,45)(18,25,26,41,28,42,43,47)(19,29,30,32,31,46,22,36)(24,37,38,40,39,48,27,44);(9,11,13,12)(10,14,15,16)(17,21,23,35)(18,26,28,43)(19,22,31,30)(20,33,34,45)(24,27,39,38)(25,41,42,47)(29,36,46,32)(37,44,48,40);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,22,21,19,23,30,35,31)(18,27,26,24,28,38,43,39)(20,32,33,46,34,36,45,29)(25,40,41,48,42,44,47,37)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fh,d^2=fi,e^2=gh,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, 6739051);