#134506

order	512
nilpotency class	4
isoclinism	isoclinic class of #134475
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,5)(4,6)(9,10)(11,23,12,27)(13,19)(14,30)(15,21)(16,22)(17,24,35,26)(18,28,34,25)(20,32)(29,39)(31,37,40,36)(33,38);(9,11,16,12)(10,17)(13,24,15,26)(14,28,33,25)(18,20)(19,23)(21,27)(22,35)(29,31,32,40)(30,36)(34,39)(37,38);(1,3)(2,5)(4,8)(6,7)(9,12,16,11)(10,18,22,34)(13,24,15,26)(14,31,33,40)(17,20,35,39)(19,37,21,36)(23,38,27,30)(25,29,28,32)

Polycyclic presentation:

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

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, 41702);