#6443197

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

5041 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)(9,10,15,20)(11,21,25,31)(12,17,29,36)(13,18,14,19)(16,24,32,22)(23,33,38,39)(26,30,37,40)(27,34,28,35);(1,2,3,4)(9,11,14,24,15,25,13,22)(10,16,18,31,20,32,19,21)(12,26,27,38,29,37,28,23)(17,33,35,40,36,39,34,30);(5,6,7,8)(11,22)(13,14)(16,21)(18,19)(23,26)(24,25)(27,28)(30,33)(31,32)(34,35)(37,38)(39,40);(9,12)(10,17)(11,23)(13,27)(14,28)(15,29)(16,30)(18,34)(19,35)(20,36)(21,33)(22,26)(24,37)(25,38)(31,39)(32,40);(9,13,15,14)(10,18,20,19)(11,22,25,24)(12,27,29,28)(16,31,32,21)(17,34,36,35)(23,26,38,37)(30,39,40,33)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^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, 6350393);