Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147593 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10965 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1616 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)(5,22,24,10)(6,25)(7,12)(8,15,30,45)(11,28)(14,42)(16,40)(17,47)(18,49)(19,51,53,33)(20,38)(21,35)(23,36,57,54)(26,41)(27,34)(29,58,59,43)(31,55,37,46)(32,50,60,64)(39,44)(48,63)(52,61,56,62);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,32)(10,13)(11,37)(12,30)(14,20)(17,28)(19,50)(21,48)(22,40)(23,55)(24,45)(25,31)(27,44)(29,34)(33,49)(35,60)(36,47)(38,59)(39,61)(41,54)(42,62)(43,52)(46,57)(51,63)(53,64)(56,58);(1,4,7,21)(2,9,12,35)(3,14,16,44)(5,19,24,53)(6,27,28,20)(8,29,30,59)(10,33,22,51)(11,38,25,34)(13,39,40,42)(15,43,45,58)(17,48,26,18)(23,56,57,52)(31,60,37,32)(36,62,54,61)(41,63,47,49)(46,64,55,50)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gh,f^c=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72267963, #72267995, #72267997, … |
| maximal normal subgroup of | #72267963, #72267995, #72267997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54790);