#144577

order	512
nilpotency class	4
isoclinism	isoclinic class of #144569
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	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

272 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

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

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, 51773);