#144789

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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 74 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)(7,9)(8,10)(11,12)(13,14,18,23)(15,24,25,39)(16,28,29,21)(17,20,31,37)(19,32,35,26)(22,38,36,30)(27,42,40,34)(33,43,44,41);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,16,26,18,25,29,32)(14,19,21,24,23,35,28,39)(17,27,30,41,31,40,38,43)(20,33,36,42,37,44,22,34);(1,3,4,2)(9,12)(14,20)(15,25)(16,29)(17,30)(19,33)(21,22)(23,37)(24,34)(27,43)(28,36)(31,38)(35,44)(39,42)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=dgi,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fi,d^c=di,e^c=eg,g^c=gi

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, 51985);