#6831179

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,32)(28,33)(29,45)(30,35)(31,36)(34,38)(37,46)(39,47)(40,48)(41,49)(42,50)(43,54)(44,51)(52,55)(53,56);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,39)(26,32,36,47)(28,40,42,43)(29,41,38,30)(33,48,50,54)(34,35,45,49)(37,52,53,44)(46,55,56,51);(2,5)(6,8)(25,28,41,52,31,42,30,44)(26,33,49,55,36,50,35,51)(27,37,29,43,39,53,38,40)(32,46,45,54,47,56,34,48);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(28,42)(33,50)(37,53)(40,43)(44,52)(46,56)(48,54)(51,55);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(25,29,30,27,31,38,41,39)(26,34,35,47,36,45,49,32)(28,43,44,37,42,40,52,53)(33,48,51,56,50,54,55,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6738375);