#6544602

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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:

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

Polycyclic presentation:

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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451798);