#6693397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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)(27,32)(28,40)(29,34)(30,33)(31,35)(36,45)(37,47)(38,46)(39,48)(41,51)(42,50)(43,52)(44,49)(53,56)(54,55);(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)(28,37)(29,42)(30,36)(31,39)(33,45)(34,50)(35,48)(38,41)(40,47)(43,53)(44,54)(46,51)(49,55)(52,56);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(27,36)(28,30)(32,45)(33,40)(37,39)(38,42)(41,44)(46,50)(47,48)(49,51)(53,54)(55,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,31,30)(26,33,35,40)(27,37,39,36)(29,41,43,44)(32,45,48,47)(34,49,52,51)(38,53,54,42)(46,50,55,56);(11,21)(15,18)(19,20)(23,24)(25,29,31,43)(26,34,35,52)(27,38,39,54)(28,41,30,44)(32,46,48,55)(33,49,40,51)(36,42,37,53)(45,50,47,56)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,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, 6600593);