#152845

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

220 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,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, 60041);