#134746

order	512
nilpotency class	4
isoclinism	isoclinic class of #134745
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

306 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,f^b=fi,e^c=ei,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41942);