#136510

order	512
nilpotency class	4
isoclinism	isoclinic class of #136509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8921
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

355 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43706);