Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1879 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10931 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,13)(4,9,20,33)(5,22)(6,27)(7,11)(8,15)(10,25)(12,24)(14,37,41,61)(16,39)(17,35)(18,38)(19,46,49,64)(21,51,48,34)(23,31)(26,52)(28,54,55,40)(29,45)(30,44)(32,58,60,50)(36,53)(42,62,43,59)(47,57,63,56);(1,3,6,16,7,18,24,17)(2,8,12,30,11,29,27,31)(4,14,21,42,20,41,48,43)(5,23,25,45,26,44,53,15)(9,28,34,56,33,55,51,57)(10,35,22,38,36,39,52,13)(19,47,50,54,49,63,58,40)(32,59,46,61,60,62,64,37);(1,4)(2,9,11,33)(3,14)(5,19)(6,21)(7,20)(8,28,29,55)(10,32,36,60)(12,34,27,51)(13,37,38,61)(15,40)(16,42)(17,43)(18,41)(22,46,52,64)(23,47)(24,48)(25,50)(26,49)(30,56,31,57)(35,59,39,62)(44,63)(45,54)(53,58)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307178, #72307219, #72307252, … |
| maximal normal subgroup of | #72307178, #72307219, #72307252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54680);