#144483

order	512
nilpotency class	4
isoclinism	isoclinic class of #144452
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51679);