#6732764

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2401 ◁ 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) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6639960);