Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14571 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×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:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,34,18,35)(4,21,24,10,26,38,25,39)(5,27,29,42,30,11,31,41)(16,36,46,50,22,32,49,43)(17,44,28,33,48,40,47,60)(23,51,55,37,56,63,54,57)(45,59,53,62,52,61,58,64);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,40)(12,34)(13,15)(14,35)(16,26)(17,29)(21,50)(23,53)(24,46)(25,49)(27,44)(30,47)(31,48)(32,39)(33,41)(37,62)(38,43)(42,60)(45,55)(51,61)(52,54)(56,58)(57,64)(59,63);(1,4,7,25,8,26,6,24)(2,10,13,21,14,39,12,38)(3,16,19,49,20,22,18,46)(5,23,29,54,30,56,31,55)(9,32,34,50,35,36,15,43)(11,37,41,51,27,57,42,63)(17,45,28,58,48,52,47,53)(33,59,40,64,60,61,44,62);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,37)(12,42)(13,41)(14,27)(15,44)(16,45)(18,47)(19,28)(20,48)(21,51)(22,52)(24,55)(25,54)(26,56)(32,59)(34,40)(35,60)(36,61)(38,63)(39,57)(43,62)(46,53)(49,58)(50,64)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297596);