#6697489

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6604685);