Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #386 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1869 |
| prime radical | #3449 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,12,28,31,8,11,29,38,5,25,15,43)(3,14,19,42,20,33,51,59,18,41,49,63,17,9,30,34)(4,21,23,39,22,35,44,55,24,10,27,36,26,37,53,61)(16,40,47,62,45,52,54,60,46,57,50,64,48,32,56,58);(1,3,15,30,5,17,29,49,8,18,28,51,7,20,6,19)(2,9,31,42,25,41,13,34,11,33,43,63,12,14,38,59)(4,16,44,54,26,48,23,47,24,46,53,56,22,45,27,50)(10,32,55,64,35,57,39,60,21,52,61,62,37,40,36,58);(1,4)(2,10)(3,16)(5,26)(6,27)(7,22)(8,24)(9,32)(11,21)(12,37)(13,39)(14,40)(15,44)(17,48)(18,46)(19,50)(20,45)(23,29)(25,35)(28,53)(30,54)(31,55)(33,52)(34,60)(36,38)(41,57)(42,64)(43,61)(47,49)(51,56)(58,59)(62,63);(2,11)(9,33)(10,21)(12,25)(13,38)(14,41)(31,43)(32,52)(34,59)(35,37)(36,39)(40,57)(42,63)(55,61)(58,60)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,f^b=fi,e^c=ei,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239242, #72249757, #72249764, … |
| maximal normal subgroup of | #72239242, #72249757, #72249764, … |
| maximal quotient of | #72239285, #72322918 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401628);