#6556106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1308 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6463302);