#6627637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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

1348 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)(9,10,14,13)(11,15,12,16)(17,18,23,28)(19,24,30,38)(20,33,35,42)(21,27,22,26)(25,36,41,34)(29,43,44,48)(31,40,32,39)(37,46,47,45);(2,5)(6,8)(9,11,14,12)(10,15,13,16)(17,19,23,30)(18,24,28,38)(20,29,35,44)(21,31,22,32)(25,37,41,47)(26,39,27,40)(33,43,42,48)(34,45,36,46);(9,10,14,13)(11,16,12,15)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,33)(27,42)(28,41)(30,44)(31,45)(32,46)(38,47)(39,43)(40,48);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,16,13,15)(19,30)(24,38)(29,44)(31,32)(37,47)(39,40)(43,48)(45,46);(9,13,14,10)(11,15,12,16)(17,21,23,22)(18,26,28,27)(19,31,30,32)(20,34,35,36)(24,39,38,40)(25,33,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534833);