#139689

order	512
nilpotency class	4
isoclinism	isoclinic class of #139558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 317 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18,22,29,24,30,23,28)(19,31,35,44,37,47,36,40)(20,27,21,26,39,45,32,46)(25,33,42,48,43,38,41,34);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(17,19,32,48,22,35,20,38,24,37,21,34,23,36,39,33)(18,25,27,31,28,41,46,40,30,43,45,47,29,42,26,44);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(17,20,23,32,24,39,22,21)(18,26,29,45,30,46,28,27)(19,33,36,34,37,38,35,48)(25,40,42,31,43,44,41,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=gi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,e^b=eh,d^c=dh,f^c=fh

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, 46885);