#6555022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

1279 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6462218);