#136219

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	C₄	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8870
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=gi,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, 43415);