#133364

order	512
nilpotency class	4
isoclinism	isoclinic class of #133362
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×Q₈	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#2431	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 815 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)(12,17,24,15)(13,26)(14,19)(16,22)(18,27)(21,37,38,36)(23,35)(25,31,29,34)(28,32)(30,39,33,40);(1,3)(2,5)(4,8)(6,7)(10,15)(12,24)(13,27)(16,30)(17,19)(18,31)(20,36)(21,38)(22,28)(26,34)(32,33)(35,37);(9,11,14,23)(10,16,19,32)(12,21,24,38)(13,28,27,22)(15,30,17,33)(18,35,26,20)(25,40,29,39)(31,37,34,36)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,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, 40560);