#6443165

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4992 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)(11,21,26,33)(12,17,30,38)(13,19,14,18)(16,25,34,24)(22,35,23,39)(27,31,40,32)(28,37,29,36);(1,2,3,4)(9,11,13,24,15,26,14,25)(10,16,18,21,20,34,19,33)(12,27,29,22,30,40,28,23)(17,35,37,31,38,39,36,32);(5,6,7,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,36)(19,37)(20,38)(21,35)(23,26)(24,27)(25,40)(32,34)(33,39);(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,32)(18,36)(19,37)(20,38)(21,39)(22,26)(24,40)(25,27)(31,34)(33,35);(9,13,15,14)(10,18,20,19)(11,24,26,25)(12,28,30,29)(16,21,34,33)(17,36,38,37)(22,27,23,40)(31,35,32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350361);