#6631389

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631114
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,30,33)(27,31,36,45)(28,38,40,48)(29,41,43,34)(32,42,47,39)(35,49,50,56)(37,53,52,46)(44,54,55,51);(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(25,27,30,36)(26,31,33,45)(28,35,40,50)(29,37,43,52)(32,44,47,55)(34,46,41,53)(38,49,48,56)(39,51,42,54);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,40)(26,32,33,47)(27,35,36,50)(29,42,43,39)(31,44,45,55)(34,48,41,38)(37,54,52,51)(46,56,53,49);(2,6)(3,7)(11,17)(15,20)(18,23)(22,24)(27,36)(31,45)(35,50)(37,52)(44,55)(46,53)(49,56)(51,54);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(26,33)(31,45)(32,47)(34,41)(38,48)(44,55)(46,53)(49,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538585);