#6833692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6740888);