#6565222

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472418);