#144787

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 ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 390 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)(15,24)(16,28)(17,20)(18,23)(19,26)(21,29)(22,30)(25,37)(27,41)(31,36)(32,40)(33,39)(34,38)(35,43)(42,44);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,16,26,18,25,29,38)(14,19,21,24,23,34,28,37)(17,27,30,40,31,39,43,42)(20,32,35,41,36,44,22,33);(1,3,4,2)(9,12)(14,20)(15,25)(16,29)(17,30)(19,32)(21,22)(23,36)(24,33)(27,42)(28,35)(31,43)(34,44)(37,41)(39,40)

Polycyclic presentation:

2^9|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, 51983);