#6567867

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fi,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, 6475063);