Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13958 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1107 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)(3,13,17,36)(4,9)(5,21,23,49)(6,11)(7,12)(8,16,27,44)(10,22,32,50)(14,37,40,34)(15,41)(18,45,47,62)(19,30)(20,31)(24,33)(25,51,28,39)(26,43)(29,46,54,63)(35,57)(38,52)(42,58)(48,55)(53,61)(56,60)(59,64);(1,3)(2,8,12,27)(4,14)(5,15)(6,16)(7,17)(9,28,31,25)(10,26,32,41)(11,13,33,36)(18,38)(19,39)(20,40)(21,35,49,58)(22,42)(23,43)(24,44)(29,53,54,52)(30,37,55,34)(45,59,62,56)(46,60)(47,61)(48,51)(50,57)(63,64);(1,4,6,19,7,20,24,48)(2,9,11,30,12,31,33,55)(3,14,16,39,17,40,44,51)(5,18,22,46,23,47,50,63)(8,25,13,34,27,28,36,37)(10,29,21,45,32,54,49,62)(15,38,42,60,43,61,57,64)(26,52,35,56,41,53,58,59);(1,5)(2,10,12,32)(3,15)(4,18)(6,22)(7,23)(8,26,27,41)(9,29,31,54)(11,21,33,49)(13,35,36,58)(14,38)(16,42)(17,43)(19,46)(20,47)(24,50)(25,52,28,53)(30,45,55,62)(34,56,37,59)(39,60)(40,61)(44,57)(48,63)(51,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285720, #72285922, #72286087, … |
| maximal normal subgroup of | #72285720, #72285922, #72286087, … |
| maximal quotient of | C2×#364058 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271254);