#144888

order	512
nilpotency class	4
isoclinism	isoclinic class of #142232
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	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 495 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=eg,f^c=fg,g^c=gi,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, 52084);