#6537362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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)(19,24)(20,31)(21,26)(22,35)(23,27)(25,32)(28,43)(29,38)(30,39)(33,41)(34,40)(36,42)(37,46)(44,47)(45,48);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,27,39)(20,28,33,45)(21,34,22,29)(25,37,42,47)(26,40,35,38)(31,43,41,48)(32,46,36,44);(10,13)(14,16)(17,20,23,33)(18,25,27,42)(19,28,30,45)(21,32,22,36)(24,37,39,47)(26,41,35,31)(29,44,34,46)(38,43,40,48);(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,27,35)(19,29,30,34)(20,32,33,36)(24,38,39,40)(25,41,42,31)(28,44,45,46)(37,43,47,48);(11,15)(14,16)(19,30)(24,39)(28,45)(29,34)(37,47)(38,40)(43,48)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,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, 6444558);