Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442590 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,22)(12,18)(13,19)(14,37)(15,20)(16,21)(17,25)(23,41)(24,45)(26,40)(27,48)(28,54)(29,42)(30,43)(31,60)(32,44)(33,61)(34,62)(35,46)(36,47)(38,51)(39,52)(49,64)(50,65)(53,69)(55,66)(56,71)(57,67)(58,72)(59,68)(63,70);(9,11,16,27)(10,17,21,40)(12,28,30,51)(13,33,36,52)(14,26,15,25)(18,41,43,65)(19,45,47,69)(20,48,37,22)(23,31,50,32)(24,34,53,35)(29,56,57,49)(38,60,54,44)(39,62,61,46)(42,66,67,63)(55,58,70,59)(64,72,71,68);(1,3,4,8)(2,5,6,7)(9,12,15,32,16,30,14,31)(10,18,20,44,21,43,37,60)(11,23,26,28,27,50,25,51)(13,29,35,59,36,57,34,58)(17,38,22,41,40,54,48,65)(19,42,46,68,47,67,62,72)(24,49,33,55,53,56,52,70)(39,63,45,64,61,66,69,71);(9,13,14,34,16,36,15,35)(10,19,37,62,21,47,20,46)(11,24,25,52,27,53,26,33)(12,29,31,58,30,57,32,59)(17,39,48,69,40,61,22,45)(18,42,60,72,43,67,44,68)(23,49,51,70,50,56,28,55)(38,63,65,71,54,66,41,64);(12,30)(18,43)(23,50)(28,51)(29,57)(31,32)(38,54)(41,65)(42,67)(44,60)(49,56)(55,70)(58,59)(63,66)(64,71)(68,72)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373281);