#144480

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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 ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 335 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24)(25,26,28,33)(27,37,39,31)(29,32,42,48)(30,35,43,36)(34,50,49,44)(38,46,41,51)(40,55,54,47)(45,53,52,56);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23)(25,27,28,39)(26,31,33,37)(29,40,42,54)(30,38,43,41)(32,47,48,55)(34,45,49,52)(35,51,36,46)(44,53,50,56);(9,12,13,19)(10,15,16,23)(11,17,18,24)(14,20,21,22)(26,32)(27,38)(29,44)(30,43)(31,45)(33,48)(34,36)(35,49)(37,52)(39,41)(40,54)(42,50)(46,55)(47,51)

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=h,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi,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, 51676);