#6436944

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,31,36)(27,32,39,48)(28,41,44,52)(29,35,30,34)(33,45,51,43)(37,53,42,56)(38,49,40,47)(46,55,50,54);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,31,39)(26,32,36,48)(28,42,44,37)(29,38,30,40)(33,50,51,46)(34,47,35,49)(41,56,52,53)(43,55,45,54);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,43)(30,45)(31,44)(32,46)(34,41)(35,52)(36,51)(38,54)(39,42)(40,55)(47,53)(48,50)(49,56);(10,15)(14,20)(16,22)(21,24)(25,29,31,30)(26,34,36,35)(27,38,39,40)(28,43,44,45)(32,47,48,49)(33,41,51,52)(37,54,42,55)(46,53,50,56);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,39)(32,48)(37,42)(38,40)(46,50)(47,49)(53,56)(54,55)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344140);