#6486483

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6393679);