#6432131

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3262 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339327);