#136221

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43417);