#126440

order	512
nilpotency class	4
isoclinism	isoclinic class of #126439
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fgi,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33636);