#6833131

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=f,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, 6740327);