#144743

order	512
nilpotency class	4
isoclinism	isoclinic class of #144674
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#2619	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 387 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)(3,6)(4,7)(5,8)(10,15)(11,19)(12,23)(16,28)(17,24)(18,29)(20,34)(25,37)(26,39)(31,38)(32,35)(36,40);(1,3,5,4)(2,6,8,7)(9,10,14,17)(11,20,19,32)(12,24,23,15)(13,26,27,18)(16,30,31,22)(21,34,33,35)(25,29,37,39)(28,36,38,40);(2,7)(3,4)(6,8)(9,11,12,21,14,19,23,33)(10,16,15,28,17,31,24,38)(13,22,25,36,27,30,37,40)(18,20,29,35,26,32,39,34)

Polycyclic presentation:

2^9|b^2=gi,c^2=d,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fi,e^c=egi,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, 51939);