#133547

order	512
nilpotency class	4
isoclinism	isoclinic class of #133329
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator, prime radical	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 82 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)(11,15)(12,22)(13,26)(14,18)(16,25)(17,28)(19,36)(20,34)(21,31)(23,40)(24,35)(27,33)(29,37)(30,39)(32,38);(2,5)(3,7)(9,11,14,21)(10,15,18,31)(12,23,24,19)(13,20,28,39)(16,32,33,29)(17,34,26,30)(22,36,35,40)(25,38,27,37);(9,12,14,24)(10,16,18,33)(11,19,21,23)(13,27,28,25)(15,29,31,32)(17,35,26,22)(20,38,39,37)(30,40,34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40743);