Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14358 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1848 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)(4,20,25,37)(5,11,28,42)(6,13)(7,12)(8,14)(9,18)(10,24,38,23)(16,44,47,31)(17,43,49,39)(19,33)(21,51,46,35)(22,53)(26,58,48,32)(27,41,29,40)(30,34)(36,55)(45,59)(50,57)(52,60)(54,63)(56,61)(62,64);(1,3,6,18,8,19,7,30)(2,9,12,15,14,34,13,33)(4,21,23,16,25,46,24,47)(5,26,27,49,28,48,29,17)(10,35,20,44,38,51,37,31)(11,39,40,58,42,43,41,32)(22,54,55,64,57,45,56,52)(36,59,53,62,61,63,50,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,20)(13,37)(14,38)(15,35)(17,45)(18,46)(19,47)(21,30)(26,52)(27,55)(28,57)(29,56)(32,59)(33,51)(34,44)(39,62)(40,53)(41,50)(42,61)(43,60)(48,64)(49,54)(58,63);(1,5)(2,11)(3,17)(4,22)(6,29)(7,27)(8,28)(9,32)(10,36)(12,41)(13,40)(14,42)(15,43)(16,45)(18,48)(19,49)(20,50)(21,52)(23,56)(24,55)(25,57)(26,30)(31,59)(33,39)(34,58)(35,60)(37,53)(38,61)(44,63)(46,64)(47,54)(51,62)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290433);