#6832051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,26)(21,27)(22,39)(23,29)(24,30)(28,35)(31,41)(32,42)(33,46)(34,44)(36,43)(37,45)(38,47)(40,48);(1,3)(2,5)(4,8)(6,7)(17,19,24,34)(18,25,30,44)(20,35,37,22)(21,32,23,40)(26,28,45,39)(27,42,29,48)(31,33,38,36)(41,46,47,43);(10,13)(14,16)(17,20,21,33,24,37,23,36)(18,26,27,46,30,45,29,43)(19,31,40,22,34,38,32,35)(25,41,48,39,44,47,42,28);(2,6)(5,7)(17,21,24,23)(18,27,30,29)(19,32,34,40)(20,36,37,33)(22,38,35,31)(25,42,44,48)(26,43,45,46)(28,41,39,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,23,31,24,35,21,38)(18,28,29,47,30,39,27,41)(19,33,32,20,34,36,40,37)(25,43,42,45,44,46,48,26)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6739247);