Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460330 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,22,16,21,15,20)(11,23,26,17,27,42,28,43)(12,29,32,46,34,18,33,47)(13,35,37,50,39,49,38,19)(24,44,30,61,53,59,54,40)(25,41,56,64,57,68,58,51)(31,48,36,45,63,70,62,60)(52,71,55,67,65,66,72,69);(2,5)(6,8)(9,11,15,28,16,27,14,26)(10,17,22,42,21,43,20,23)(12,30,33,24,34,54,32,53)(13,25,38,58,39,57,37,56)(18,44,47,61,29,59,46,40)(19,41,35,64,50,68,49,51)(31,55,62,52,63,72,36,65)(45,69,70,71,60,67,48,66);(3,7)(6,8)(9,12,16,34)(10,18,21,29)(11,24,27,53)(13,36,39,62)(14,33,15,32)(17,40,43,61)(19,48,50,70)(20,47,22,46)(23,44,42,59)(25,55,57,72)(26,54,28,30)(31,38,63,37)(35,60,49,45)(41,67,68,69)(51,66,64,71)(52,58,65,56);(1,3,4,7)(2,6,5,8)(9,13,15,38,16,39,14,37)(10,19,22,35,21,50,20,49)(11,25,28,58,27,57,26,56)(12,31,33,62,34,63,32,36)(17,41,42,64,43,68,23,51)(18,45,47,70,29,60,46,48)(24,52,54,72,53,65,30,55)(40,66,44,69,61,71,59,67)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377819);