#512790

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₁₆	C₂³
commutator, socle	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#41	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=e,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ci

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419986);