#6832690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

1296 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)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,44)(32,38)(33,41)(34,39)(35,40)(36,43)(37,45)(42,46)(47,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,33,32,23)(21,34,35,46)(25,28,38,41)(27,42,40,39)(30,36,45,47)(37,43,44,48);(2,5)(6,8)(10,14)(11,16)(17,20,23,19,22,32,33,31)(18,25,28,29,26,38,41,24)(21,34,36,30,35,46,47,45)(27,39,43,44,40,42,48,37);(10,14)(13,15)(18,26)(21,35)(24,29)(25,38)(28,41)(30,45)(34,46)(36,47);(1,3,4,7)(2,6,5,8)(17,21,33,47,22,35,23,36)(18,27,41,48,26,40,28,43)(19,30,20,34,31,45,32,46)(24,37,38,42,29,44,25,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6739886);