#139017

order	512
nilpotency class	4
isoclinism	isoclinic class of #139013
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#442
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 298 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=gh,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eghi,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46213);