#152202

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

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	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 352 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286618, #72287227, #72287229
maximal normal subgroup of	#72286618, #72287227, #72287229

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59398);