#144590

order	512
nilpotency class	4
isoclinism	isoclinic class of #144587
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	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 271 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:

(3,10,12,13)(7,11,15,16)(19,27,29,43)(20,31)(22,35)(23,37,28,45)(24,39)(26,41)(30,33,36,44)(32,46)(34,47)(38,40,42,48);(1,2)(3,11,12,16)(4,14)(5,8)(6,9)(7,13,15,10)(17,18,21,25)(19,28)(20,32,34,24)(22,26,35,41)(23,29)(27,37)(30,42)(31,46,47,39)(33,40)(36,38)(43,45)(44,48);(1,3)(2,7)(4,13)(5,10)(6,12)(8,16)(9,15)(11,14)(17,19,21,29)(18,23,25,28)(20,33,34,44)(22,36,35,30)(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=i,e^2=ghi,f^2=gi,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=eg,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, 51786);