#6693455

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1158 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6600651);