#6432135

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)(13,18)(14,19)(15,20)(16,24)(17,28)(22,37)(23,33)(25,32)(27,34)(29,36)(30,35)(31,38)(39,40);(5,6)(7,8)(9,11,13,24,15,23,14,25)(10,16,19,21,20,32,18,33)(12,27,29,22,30,39,28,38)(17,34,35,31,36,40,26,37);(2,4)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,26)(20,36)(21,34)(23,38)(24,27)(25,39)(32,37)(33,40);(11,23)(16,32)(21,33)(22,38)(24,25)(27,39)(31,37)(34,40);(9,13,15,14)(10,18,20,19)(11,24,23,25)(12,28,30,29)(16,33,32,21)(17,35,36,26)(22,27,38,39)(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,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=di,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, 6339331);