#144485

order	512
nilpotency class	4
isoclinism	isoclinic class of #144454
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	C₂×D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

321 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51681);