#152486

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂⁵

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	C₂²×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 362 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) = 256

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

(2,4)(5,6)(7,8)(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);(1,2,3,4)(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);(6,8)

Polycyclic presentation:

2^9|a^2=f,b^2=df,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.

Relationships

maximal subgroup of	#72322704, #72322860, #72322872, …
maximal normal subgroup of	#72322704, #72322860, #72322872, …
maximal quotient of	#72322722, #72322725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59682);