#6832039

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

1464 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)(19,24)(20,25)(21,34)(22,27)(23,28)(26,35)(29,37)(30,38)(31,39)(32,40)(33,42)(36,43)(41,46)(44,47)(45,48);(1,3)(2,5)(4,8)(6,7)(17,19,23,30)(18,24,28,38)(20,31,32,46)(21,22,35,36)(25,39,40,41)(26,43,34,27)(29,44,45,33)(37,47,48,42);(10,13)(14,16)(17,20,22,33,23,32,36,44)(18,25,27,42,28,40,43,47)(19,29,21,31,30,45,35,46)(24,37,34,39,38,48,26,41);(2,6)(5,7)(20,32)(25,40)(29,45)(31,46)(33,44)(37,48)(39,41)(42,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,30,23,35,36,19)(18,26,27,24,28,34,43,38)(20,31,33,45,32,46,44,29)(25,41,42,37,40,39,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,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, 6739235);