#6631376

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi

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, 6538572);