Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1898 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1946 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)(3,15,20,44)(4,21,24,41)(5,11)(6,26,16,12)(7,13)(8,14)(9,19)(10,22,36,31)(17,45)(18,43,30,58)(23,55)(25,39,29,37)(27,34)(28,53)(32,59,62,64)(33,50)(35,54)(38,57)(40,56)(42,46)(47,52,51,61)(48,60)(49,63);(1,3,16,19,8,20,6,27)(2,9,26,44,14,34,12,15)(4,17,23,51,24,49,28,47)(5,18,29,50,7,30,25,56)(10,32,35,60,36,62,38,42)(11,33,39,58,13,40,37,43)(21,52,53,45,41,61,55,63)(22,46,54,64,31,48,57,59);(1,4)(2,10)(3,17)(5,22)(6,28)(7,31)(8,24)(9,32)(11,21)(12,38)(13,41)(14,36)(15,42)(16,23)(18,46)(19,51)(20,49)(25,57)(26,35)(27,47)(29,54)(30,48)(33,52)(34,62)(37,55)(39,53)(40,61)(43,63)(44,60)(45,58)(50,64)(56,59);(1,5,8,7)(2,11,14,13)(3,18,20,30)(4,22,24,31)(6,25,16,29)(9,33,34,40)(10,21,36,41)(12,37,26,39)(15,43,44,58)(17,46,49,48)(19,50,27,56)(23,54,28,57)(32,52,62,61)(35,53,38,55)(42,63,60,45)(47,59,51,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303731, #72303735, #72303747, … |
| maximal normal subgroup of | #72303731, #72303735, #72303747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401944);