#144393

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
prime radical	#3485	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 846 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,17)(13,30)(14,19)(15,20)(16,23)(18,32)(22,52)(24,44)(25,51)(26,57)(27,45)(28,42)(29,43)(31,39)(33,62)(34,67)(35,48)(36,49)(37,53)(38,55)(40,58)(41,60)(46,65)(47,64)(50,54)(56,70)(59,69)(61,72)(63,68)(66,71);(5,8)(9,11,15,24)(10,16,20,39)(12,26,29,56)(13,31,32,23)(14,34,36,25)(17,40,43,69)(18,44,30,21)(19,47,49,38)(22,35,54,28)(27,59,60,58)(33,55,65,64)(37,48,68,42)(41,70,45,57)(46,51,62,67)(50,72,52,71)(53,61,63,66);(1,3)(2,5)(4,7)(6,8)(9,12,15,29)(10,17,20,43)(11,22,24,54)(13,27,32,60)(14,35,36,28)(16,37,39,68)(18,45,30,41)(19,48,49,42)(21,50,44,52)(23,53,31,63)(25,56,34,26)(33,66,65,61)(38,69,47,40)(46,71,62,72)(51,57,67,70)(55,59,64,58)

Polycyclic presentation:

2^9|b^2=i,c^2=i,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, 51589);