#6831211

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

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

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,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6738407);