#6537957

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, 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

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,6)(3,7,8,5)(25,26)(27,32)(28,40)(29,34)(30,43)(31,35)(33,41)(36,50)(37,46)(38,47)(39,53)(42,48)(44,49)(45,52)(51,56)(54,55);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27,31,37)(26,32,35,46)(28,36,42,51)(29,38,30,39)(33,45,49,55)(34,47,43,53)(40,50,48,56)(41,52,44,54);(10,14)(15,21)(16,22)(23,24)(25,28,31,42)(26,33,35,49)(27,36,37,51)(29,41,30,44)(32,45,46,55)(34,48,43,40)(38,52,39,54)(47,56,53,50);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(27,37)(32,46)(36,51)(38,39)(45,55)(47,53)(50,56)(52,54);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,31,30)(26,34,35,43)(27,38,37,39)(28,41,42,44)(32,47,46,53)(33,48,49,40)(36,52,51,54)(45,56,55,50)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6445153);