#144676

order	512
nilpotency class	4
isoclinism	isoclinic class of #144674
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	C₂³×D₄	D₄
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 463 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)(9,10,14,19)(11,20,23,16)(12,15,24,17)(13,26,27,18)(21,35,37,32)(22,31,39,28)(25,33,40,29)(30,36,34,38);(10,15)(11,21)(12,24)(13,27)(16,30)(17,19)(18,33)(20,34)(22,38)(23,37)(26,29)(28,32)(31,35)(36,39);(2,5)(3,8)(6,7)(9,11,12,22,14,23,24,39)(10,16,17,31,19,20,15,28)(13,21,25,36,27,37,40,38)(18,32,33,34,26,35,29,30)

Polycyclic presentation:

2^9|a^2=i,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, 51872);