#126088

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	C₂³	#447
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 297 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=g,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=ehi,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, 33284);