#143763

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28233	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50959);