#144594

order	512
nilpotency class	4
isoclinism	isoclinic class of #144591
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 ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 322 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)(3,9,11,15)(4,7)(5,8)(6,12,10,13)(14,16)(17,18,20,23)(19,27,28,45)(21,33,31,46)(22,36,32,26)(24,40,38,29)(25,42,39,35)(30,47,37,43)(34,44,48,41);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,15,13)(14,16)(18,23)(24,38)(25,39)(26,36)(27,45)(33,46)(41,44)(43,47);(1,3)(2,6)(4,13)(5,11)(7,15)(8,10)(9,16)(12,14)(17,19,20,28)(18,24,23,38)(21,34,31,48)(22,37,32,30)(25,43,39,47)(26,44,36,41)(27,46,45,33)(29,42,40,35)

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=i,e^2=ghi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fg,h^a=hi,c^b=cf,f^b=fg,d^c=dg,e^c=egh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51790);