Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2058 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11323 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,12,7,29)(3,14,18,41,17,40,28,42)(4,10,22,36,21,37,23,54)(5,24,25,38,27,55,26,11)(9,30,33,16,32,48,34,47)(15,43,46,63,19,39,50,62)(20,51,52,61,44,64,53,35)(31,57,59,49,58,56,60,45);(1,3,5,16,7,18,26,48,8,17,27,47,6,28,25,30)(2,9,11,14,13,34,24,42,12,32,38,40,29,33,55,41)(4,19,20,49,23,50,53,56,21,15,44,45,22,46,52,57)(10,31,35,39,36,60,51,63,37,58,61,43,54,59,64,62);(1,4)(2,10)(3,15,17,19)(5,20)(6,22)(7,23)(8,21)(9,31,32,58)(11,35)(12,37)(13,36)(14,39,40,43)(16,45,47,49)(18,46,28,50)(24,51)(25,52)(26,53)(27,44)(29,54)(30,56,48,57)(33,59,34,60)(38,61)(41,62,42,63)(55,64)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307179, #72307228, #72307295, … |
| maximal normal subgroup of | #72307179, #72307228, #72307295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56397);