#6868326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775522);