#143798

order	512
nilpotency class	4
isoclinism	isoclinic class of #143798
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10235
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28249	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 382 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fi,g^b=gi,d^c=di,e^c=ehi,f^c=fi,h^c=hi

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, 50994);