#6868259

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

1496 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)(17,18,21,25,22,26,36,43)(19,27,30,44,29,23,37,39)(20,31,35,42,34,41,33,24)(28,45,47,38,32,48,46,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,29)(18,23,26,27)(20,32,34,28)(21,30,36,37)(24,40,42,38)(25,39,43,44)(31,45,41,48)(33,47,35,46);(10,14)(15,16)(17,20)(18,24)(19,28)(21,33)(22,34)(23,38)(25,41)(26,42)(27,40)(29,32)(30,46)(31,43)(35,36)(37,47)(39,45)(44,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,29)(23,27)(28,32)(30,37)(38,40)(39,44)(45,48)(46,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,36)(18,25,26,43)(19,30,29,37)(20,33,34,35)(23,39,27,44)(24,41,42,31)(28,46,32,47)(38,45,40,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6775455);