#144670

order	512
nilpotency class	4
isoclinism	isoclinic class of #144670
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#29721	C₂
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 484 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=g,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,c^b=cf,d^b=di,e^b=eh,e^c=egi,f^c=fi,h^c=hi,f^e=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, 51866);