#6859909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6767105);