#133438

order	512
nilpotency class	4
isoclinism	isoclinic class of #133434
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2310 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

606 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,10,14,19,15,20,11,21)(12,24,27,35,26,17,22,33)(13,18,23,31,29,34,28,16)(25,38,36,30,40,32,39,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,20,31)(12,22,26,27)(13,28,29,23)(17,30,24,37)(18,19,34,21)(25,39,40,36)(32,33,38,35);(1,3)(2,5)(4,7)(6,8)(9,12,15,26)(10,17)(11,22,14,27)(13,25,29,40)(16,30)(18,32)(19,33)(20,24)(21,35)(23,36,28,39)(31,37)(34,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 40634);