#131747

order	512
nilpotency class	4
isoclinism	isoclinic class of #131677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62264	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 564 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=egi,f^c=fg

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38943);