#144164

order	512
nilpotency class	4
isoclinism	isoclinic class of #144131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fghi,h^b=hi,d^c=dg,e^c=eh,f^c=fghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51360);