#135385

order	512
nilpotency class	4
isoclinism	isoclinic class of #124851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=dg,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42581);