#133573

order	512
nilpotency class	4
isoclinism	isoclinic class of #133394
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	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#560	C₄
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,d^c=dh,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, 40769);