#133584

order	512
nilpotency class	4
isoclinism	isoclinic class of #133350
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

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

normal subgroups only

This group has 694 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)(11,15)(12,22,23,32)(13,17)(14,18)(16,24,30,25)(19,34,35,39)(20,28)(21,29)(26,33)(27,36,40,37)(31,38);(2,5)(3,7)(9,11,14,21)(10,15,18,29)(12,19,23,35)(13,20,26,38)(16,27,30,40)(17,31,33,28)(22,39,32,34)(24,36,25,37);(9,12)(10,16)(11,19)(13,25)(14,23)(15,27)(17,32)(18,30)(20,37)(21,35)(22,33)(24,26)(28,39)(29,40)(31,34)(36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=hi,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fi,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, 40780);