#152806

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator	C₂²	C₂²×C₃₂
socle	C₂³	C₂²×C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 60002);