#6860660

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1232 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:

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

Polycyclic presentation:

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

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, 6767856);