#137530

order	512
nilpotency class	4
isoclinism	isoclinic class of #137493
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	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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)(17,18,22,26,23,28,20,24)(19,29,32,44,33,47,30,46)(21,35,37,27,38,43,34,45)(25,40,41,48,42,36,39,31);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(18,24,28,26)(25,39,42,41)(27,35,45,43)(29,46,47,44);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,19)(18,25)(20,30)(21,36)(22,32)(23,33)(24,39)(26,41)(27,44)(28,42)(29,35)(31,37)(34,48)(38,40)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gi,e^2=ghi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44726);