#144390

order	512
nilpotency class	4
isoclinism	isoclinic class of #144390
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 56 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)(4,6)(7,8)(9,10)(11,19)(12,24)(13,17)(14,18)(15,21)(16,26)(20,36)(22,32)(23,35)(25,31)(27,39)(28,34)(29,37)(30,38)(33,40);(5,8)(9,11)(10,15)(12,25)(13,20)(14,22)(16,32)(17,29)(18,31)(19,24)(21,26)(23,28)(27,38)(30,34)(33,35)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(11,20)(13,28)(15,29)(16,24)(17,34)(19,35)(21,37)(22,23)(25,38)(27,40)(30,31)(32,36)

Polycyclic presentation:

2^9|d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,d^c=dg,f^c=fi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51586);