#6832768

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fhi,d^2=i,e^2=gh,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, 6739964);