Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3243 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)(4,18,21,33)(5,22,24,10)(6,11)(7,12)(8,16,28,42)(9,20,31,25)(14,38,39,61)(15,36)(17,37)(19,48)(23,52,54,32)(26,34)(27,55)(29,44,57,63)(30,49)(35,41,59,45)(40,58)(43,60)(46,62)(47,50)(51,53)(56,64);(1,3)(2,8)(4,14)(5,16)(6,17)(7,15)(9,27)(10,13)(11,29)(12,28)(18,46)(19,40)(20,41)(21,39)(22,36)(23,44)(24,42)(25,45)(26,43)(30,35)(31,56)(32,37)(33,58)(34,57)(38,47)(48,61)(49,55)(50,62)(51,59)(52,60)(53,64)(54,63);(1,4,7,21)(2,9,12,31)(3,14,15,39)(5,19,24,50)(6,25,26,20)(8,27,28,56)(10,30,22,51)(11,33,34,18)(13,35,36,59)(16,40,42,62)(17,45,43,41)(23,53,54,49)(29,58,57,46)(32,48,52,47)(37,61,60,38)(44,64,63,55);(3,15)(8,28)(13,36)(14,39)(16,42)(17,43)(27,56)(29,57)(35,59)(37,60)(38,61)(40,62)(41,45)(44,63)(46,58)(55,64)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303033, #72303730, #72303748, … |
| maximal normal subgroup of | #72303033, #72303730, #72303748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400608);