#134505

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 418 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,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, 41701);