#133329

order	512
nilpotency class	4
isoclinism	isoclinic class of #133329
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#46276	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 858 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) = 64

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

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

Polycyclic presentation:

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

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, 40525);