#6819495

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,22)(19,24,29,38)(20,25,33,31)(21,26,37,36)(27,30,39,43)(28,34,41,48)(32,40,35,42)(44,46,45,47);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,34)(22,38)(23,29)(25,39)(26,41)(27,31)(28,36)(32,45)(33,43)(35,44)(37,48)(40,47)(42,46);(9,12,13,11)(10,14,15,16)(17,20,22,31,23,33,18,25)(19,27,24,30,29,39,38,43)(21,35,36,40,37,32,26,42)(28,45,34,47,41,44,48,46);(2,5)(6,8)(17,18,23,22)(19,24,29,38)(20,31,33,25)(21,26,37,36)(27,43,39,30)(28,34,41,48)(32,42,35,40)(44,47,45,46);(1,3,4,7)(2,6,5,8)(17,21,22,36,23,37,18,26)(19,28,24,34,29,41,38,48)(20,32,31,42,33,35,25,40)(27,44,30,46,39,45,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6726691);