#125744

order	512
nilpotency class	4
isoclinism	isoclinic class of #125709
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	#3150	C₂²
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=fi,e^2=hi,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=dfi,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, 32940);