#144163

order	512
nilpotency class	4
isoclinism	isoclinic class of #144122
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2011
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 256

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

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=ehi,f^b=fghi,h^b=hi,d^c=dgi,e^c=eh,f^c=fghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51359);