Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1864 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1864 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1025 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)(3,16,21,39)(4,22,25,57)(5,12)(6,28)(7,31)(8,34)(9,15)(10,19,40,20)(11,24,43,58)(13,27)(14,26)(17,50)(18,47,35,45)(23,48,36,46)(29,41,32,38)(30,44,33,42)(37,54)(49,59)(51,63)(52,62)(53,60)(55,64)(56,61);(1,3,8,35,9,21,5,18)(2,10,12,38,15,40,34,41)(4,17,36,53,25,54,23,51)(6,29,27,19,7,32,26,20)(11,37,42,60,43,50,44,63)(13,45,28,16,14,47,31,39)(22,49,48,64,57,62,46,61)(24,55,33,59,58,56,30,52);(1,4)(2,11)(3,17)(5,23)(6,30)(7,33)(8,36)(9,25)(10,37)(12,42)(13,46)(14,48)(15,43)(16,49)(18,51)(19,55)(20,56)(21,54)(22,28)(24,27)(26,58)(29,52)(31,57)(32,59)(34,44)(35,53)(38,60)(39,62)(40,50)(41,63)(45,61)(47,64);(1,5,9,8)(2,12,15,34)(3,18,21,35)(4,23,25,36)(6,26,7,27)(10,38,40,41)(11,42,43,44)(13,28,14,31)(16,47,39,45)(17,51,54,53)(19,29,20,32)(22,48,57,46)(24,30,58,33)(37,60,50,63)(49,64,62,61)(52,56,59,55)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285805, #72285848, #72286096, … |
| maximal normal subgroup of | #72285805, #72285848, #72286096, … |
| maximal quotient of | #72310176, #72321677, #72322954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401515);