#6557780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1282 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464976);