#145384

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
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 ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 378 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)(7,9)(8,10)(11,12)(14,15)(16,27)(17,28)(18,19)(21,23)(22,25)(24,26);(1,2)(3,4)(6,9)(10,11)(13,14,20,24)(15,17,26,27)(16,21,28,25)(18,23,19,22);(2,4)(6,10)(8,12)(9,11)(13,15,17,24,18,21,16,23,20,26,27,14,19,25,28,22)

Polycyclic presentation:

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

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, 52580);