#133467

order	512
nilpotency class	4
isoclinism	isoclinic class of #133338
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	C₂³×Q₈	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 637 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,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, 40663);