#144890

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
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	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(2,3)(5,6)(7,10)(8,11)(9,12)(13,14,19,24)(15,20,29,41)(16,21,33,42)(17,34,31,22)(18,35,32,23)(25,43,26,39)(27,44,30,38)(28,37,36,40);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,30)(17,25)(18,36)(19,29)(21,38)(22,39)(23,37)(24,41)(26,31)(27,33)(28,32)(34,43)(35,40)(42,44);(1,3,4,2)(6,10)(7,8)(11,12)(14,21)(15,25)(16,31)(17,33)(18,32)(20,37)(22,23)(24,42)(26,29)(27,36)(28,30)(34,35)(39,43)(40,41)

Polycyclic presentation:

2^9|a^2=i,c^2=d,e^2=gi,f^2=g,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=fgi,g^b=gi,h^b=hi,e^c=eg,f^c=fgi,g^c=gi,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, 52086);