#152379

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₄⋊C₁₆
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=d,d^2=i,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59575);