#6832542

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1291 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,31)(21,35)(22,27)(23,28)(25,33)(26,36)(29,44)(30,39)(32,41)(34,47)(37,43)(38,46)(40,48)(42,45);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,39)(20,32,33,48)(21,23,36,37)(25,40,31,41)(26,43,35,28)(29,45,46,34)(38,47,44,42);(2,5)(6,8)(10,14)(11,16)(17,20,23,34,22,33,37,45)(18,25,28,42,27,31,43,47)(19,29,21,32,30,46,36,48)(24,38,35,40,39,44,26,41);(10,14)(13,15)(20,33)(25,31)(29,46)(32,48)(34,45)(38,44)(40,41)(42,47);(1,3,4,7)(2,6,5,8)(17,21,23,30,22,36,37,19)(18,26,28,24,27,35,43,39)(20,32,34,46,33,48,45,29)(25,41,42,38,31,40,47,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fh,e^2=ghi,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, 6739738);