#6868006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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

1494 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)(17,18,21,26,22,25,36,42)(19,27,31,38,29,23,30,39)(20,32,35,40,34,41,33,24)(28,44,46,37,45,48,47,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,29)(18,23,25,27)(20,28,34,45)(21,31,36,30)(24,37,40,43)(26,39,42,38)(32,48,41,44)(33,47,35,46);(10,14)(15,16)(17,20,22,34)(18,24,25,40)(19,28,29,45)(21,33,36,35)(23,37,27,43)(26,41,42,32)(30,46,31,47)(38,48,39,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,25)(19,29)(24,40)(26,42)(28,45)(30,31)(32,41)(46,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,36)(18,26,25,42)(19,30,29,31)(20,33,34,35)(23,38,27,39)(24,41,40,32)(28,46,45,47)(37,48,43,44)

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=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775202);