#131729

order	512
nilpotency class	4
isoclinism	isoclinic class of #131660
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	#62332	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 572 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)(17,18,23,30,25,29,21,40,24,31,42,48,22,43,41,28)(19,32,35,26,34,45,33,47,20,36,39,46,38,27,37,44);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,24,20)(18,26,31,46)(21,39,41,35)(22,34,25,38)(23,37,42,33)(27,28,45,40)(29,44,43,47)(30,32,48,36);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,20,24,19)(18,27,31,45)(21,37,41,33)(22,34,25,38)(23,39,42,35)(26,28,46,40)(29,32,43,36)(30,44,48,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=eg,f^c=fgi

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, 38925);