#131101

order	512
nilpotency class	4
isoclinism	isoclinic class of #130734
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#61251	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 560 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:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(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);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(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);(2,7)(4,13)(6,9)(10,12)(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=g,d^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38297);