#137531

order	512
nilpotency class	4
isoclinism	isoclinic class of #137497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 315 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gi,e^2=gh,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,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, 44727);