#6867655

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(3,7)(4,6)(5,8)(17,18,23,36,22,27,19,24)(20,30,28,25,33,38,32,41)(21,34,37,39,35,26,29,43)(31,45,42,48,46,44,47,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,36)(20,28,33,32)(21,29,35,37)(25,38,41,30)(26,39,34,43)(31,42,46,47)(40,45,48,44);(10,14)(15,16)(17,20,22,33)(18,25,27,41)(19,28,23,32)(21,31,35,46)(24,38,36,30)(26,40,34,48)(29,42,37,47)(39,45,43,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,35)(18,26,27,34)(19,29,23,37)(20,31,33,46)(24,39,36,43)(25,40,41,48)(28,42,32,47)(30,44,38,45);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(21,35)(26,34)(29,37)(31,46)(39,43)(40,48)(42,47)(44,45)

Polycyclic presentation:

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

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, 6774851);