#144720

order	512
nilpotency class	4
isoclinism	isoclinic class of #144720
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#29748	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,5)(4,6)(7,8)(10,16)(11,21)(12,14)(13,27)(17,32)(18,35)(19,20)(22,40)(23,28)(24,38)(25,39)(26,29)(30,33)(34,36);(1,3)(2,5)(4,7)(6,8)(9,10)(11,22)(12,19)(13,18)(14,16)(15,20)(17,23)(21,36)(24,34)(25,30)(26,37)(27,31)(28,40)(29,35)(32,39)(33,38);(5,8)(9,11)(10,17,20,34)(12,24)(13,28)(14,23)(15,25)(16,30,19,22)(18,36,35,32)(21,27)(26,39)(29,38)(31,33,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51916);