#125772

order	512
nilpotency class	4
isoclinism	isoclinic class of #125709
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

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

normal subgroups only

This group has 286 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)(11,22)(12,21)(13,17)(14,20)(15,18)(16,23)(19,28)(24,31)(25,33)(26,32)(27,34)(29,35)(30,38)(36,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,25)(10,16)(12,23,15,26)(13,24,29,39)(17,30)(18,22)(19,36)(20,32)(21,33)(27,37,28,38)(31,34)(35,40);(1,3,5,4)(2,6,7,8)(10,17)(12,15)(13,28)(16,30)(18,34)(19,21)(20,35)(22,31)(23,26)(24,38)(27,29)(32,40)(33,36)(37,39)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,d^c=dg,e^c=eh,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, 32968);