#139691

order	512
nilpotency class	4
isoclinism	isoclinic class of #139562
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	C₂²×D₈	C₂×C₄
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

350 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fi,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, 46887);