#6443190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350386);