#6537269

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18,23,26)(19,24,30,38)(20,31,33,42)(21,27,22,35)(25,36,41,32)(28,43,46,48)(29,39,34,40)(37,44,47,45);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,28)(21,34)(22,29)(23,30)(25,37)(26,38)(27,40)(31,43)(32,45)(33,46)(35,39)(36,44)(41,47)(42,48);(10,13)(14,16)(17,20,23,33)(18,25,26,41)(19,28,30,46)(21,32,22,36)(24,37,38,47)(27,42,35,31)(29,44,34,45)(39,43,40,48);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(18,26)(24,38)(25,41)(27,35)(31,42)(37,47)(39,40)(43,48);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,27,26,35)(19,29,30,34)(20,32,33,36)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

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, 6444465);