#145930

order	512
nilpotency class	4
isoclinism	isoclinic class of #143935
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53126);