#6550456

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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)(3,6)(4,5)(7,8)(25,26,30,35,31,36,29,34)(27,32,38,46,40,47,39,50)(28,41,42,49,43,48,44,33)(37,51,53,56,54,55,52,45);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27)(26,32)(28,37)(29,39)(30,38)(31,40)(33,45)(34,50)(35,46)(36,47)(41,51)(42,53)(43,54)(44,52)(48,55)(49,56);(1,3,4,7)(2,6,5,8)(10,14)(15,21)(16,22)(23,24)(25,28,31,43)(26,33,36,49)(27,37,40,54)(29,42,30,44)(32,45,47,56)(34,41,35,48)(38,52,39,53)(46,55,50,51);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,29,31,30)(26,34,36,35)(27,38,40,39)(28,42,43,44)(32,46,47,50)(33,41,49,48)(37,52,54,53)(45,55,56,51);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,31,30)(26,35,36,34)(27,39,40,38)(28,42,43,44)(32,46,47,50)(33,48,49,41)(37,53,54,52)(45,55,56,51)

Polycyclic presentation:

2^9|a^2=g,c^2=fi,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,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, 6457652);