#6690083

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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1162 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,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, 6597279);