#6693423

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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 ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 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)(25,26,31,30)(27,32,37,35)(28,29,33,39)(34,36,40,38);(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)(28,38)(29,34)(30,35)(31,37)(33,36)(39,40);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,29)(27,34)(30,39)(31,33)(32,36)(35,38)(37,40);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,26,31,30)(27,35,37,32)(28,29,33,39)(34,38,40,36);(11,21)(15,18)(19,20)(23,24)(25,29,31,39)(26,33,30,28)(27,36,37,38)(32,40,35,34)

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,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, 6600619);