#143690

order	512
nilpotency class	4
isoclinism	isoclinic class of #143259
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1624
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10228
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 443 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50886);