#131726

order	512
nilpotency class	4
isoclinism	isoclinic class of #131660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#62328	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

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

normal subgroups only

This group has 608 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)(17,18,20,25,34,42,48,37,22,26,35,38,21,27,33,23)(19,28,30,39,46,41,45,43,32,24,29,40,31,44,47,36);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,23)(20,33)(21,34)(24,36)(25,27)(26,37)(28,43)(29,45)(30,47)(31,46)(35,48)(38,42)(39,41)(40,44);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,19)(18,24)(20,29)(21,31)(22,32)(23,36)(25,39)(26,28)(27,41)(30,35)(33,45)(34,46)(37,43)(38,40)(42,44)(47,48)

Polycyclic presentation:

2^9|a^2=f,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, 38922);