Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6)(7,19)(8,25)(9,15)(10,16)(11,17)(12,18)(13,23)(14,28)(20,47)(21,43)(22,38)(24,37)(26,39)(27,53)(29,42)(30,41)(31,36)(32,44)(33,45)(34,46)(35,50)(40,55)(48,65)(49,59)(51,61)(52,60)(54,62)(56,64)(57,63)(58,67)(66,68);(1,2,3,4)(5,7,10,23,12,24,11,22)(6,13,17,19,18,38,16,37)(8,26,29,20,30,49,28,50)(9,31,33,21,34,51,32,52)(14,39,41,35,42,59,25,47)(15,43,44,36,46,60,45,61)(27,48,55,54,57,67,56,66)(40,58,53,62,64,65,63,68);(5,8,11,29,12,30,10,28)(6,14,16,41,18,42,17,25)(7,20,22,49,24,50,23,26)(9,27,33,56,34,57,32,55)(13,35,37,59,38,47,19,39)(15,40,44,63,46,64,45,53)(21,48,51,66,52,67,31,54)(36,58,60,68,61,65,43,62);(5,9)(6,15)(7,21)(8,27)(10,32)(11,33)(12,34)(13,36)(14,40)(16,44)(17,45)(18,46)(19,43)(20,48)(22,51)(23,31)(24,52)(25,53)(26,54)(28,55)(29,56)(30,57)(35,58)(37,60)(38,61)(39,62)(41,63)(42,64)(47,65)(49,66)(50,67)(59,68);(5,10,12,11)(6,16,18,17)(7,22,24,23)(8,28,30,29)(9,32,34,33)(13,19,38,37)(14,41,42,25)(15,44,46,45)(20,49,50,26)(21,51,52,31)(27,55,57,56)(35,39,47,59)(36,43,61,60)(40,63,64,53)(48,66,67,54)(58,62,65,68)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350486);