Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,4,3)(5,6)(7,18)(8,24)(9,14)(10,15)(11,17)(12,21)(13,29)(16,26)(19,38)(20,48)(22,37)(23,44)(25,47)(27,54)(28,45)(30,40)(31,59)(32,42)(33,46)(34,43)(35,52)(36,51)(39,57)(41,53)(49,62)(50,61)(55,65)(56,64)(58,63)(60,67)(66,68);(2,3)(5,7)(6,12)(8,25)(9,31)(10,21)(11,22)(13,38)(14,41)(15,18)(16,37)(17,44)(19,30)(20,34)(23,26)(24,35)(27,49)(28,52)(29,50)(32,53)(33,51)(36,43)(39,60)(40,61)(42,59)(45,47)(46,48)(54,68)(55,57)(56,66)(58,67)(62,64)(63,65);(1,2,4,3)(5,8,26,30,10,28,11,29)(6,13,17,45,15,40,16,24)(7,19,23,52,21,50,22,25)(9,27,34,58,32,56,33,57)(12,35,44,38,18,47,37,61)(14,39,46,64,42,63,43,54)(20,49,53,67,51,66,31,55)(36,60,41,62,48,65,59,68);(1,3,4,2)(5,9,10,32)(6,14,15,42)(7,20,21,51)(8,27,28,56)(11,33,26,34)(12,36,18,48)(13,39,40,63)(16,43,17,46)(19,49,50,66)(22,31,23,53)(24,54,45,64)(25,55,52,67)(29,57,30,58)(35,60,47,65)(37,59,44,41)(38,62,61,68);(8,28)(13,40)(19,50)(24,45)(25,52)(27,56)(29,30)(35,47)(38,61)(39,63)(49,66)(54,64)(55,67)(57,58)(60,65)(62,68)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351502);