Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14336 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2083 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)(3,8)(4,17,20,9)(5,10)(6,12)(7,11)(13,38,18,27)(14,28)(15,29)(16,30)(19,33,22,31)(21,46,49,32)(23,35)(24,34)(25,37)(26,36)(39,53,41,51)(40,60,47,52)(42,54)(43,55)(44,56)(45,57)(48,59,50,58)(61,64,62,63);(1,3)(2,8)(4,18,20,13)(5,14)(6,16)(7,15)(9,27,17,38)(10,28)(11,29)(12,30)(19,41,22,39)(21,47,49,40)(23,43)(24,42)(25,45)(26,44)(31,51,33,53)(32,52,46,60)(34,54)(35,55)(36,56)(37,57)(48,62,50,61)(58,63,59,64);(1,4,6,21,7,20,26,49)(2,9,12,32,11,17,36,46)(3,13,16,40,15,18,44,47)(5,22,23,50,24,19,25,48)(8,27,30,52,29,38,56,60)(10,33,35,59,34,31,37,58)(14,41,43,62,42,39,45,61)(28,53,55,64,54,51,57,63);(1,5,7,24)(2,10,11,34)(3,14,15,42)(4,19,20,22)(6,25,26,23)(8,28,29,54)(9,31,17,33)(12,37,36,35)(13,39,18,41)(16,45,44,43)(21,50,49,48)(27,51,38,53)(30,57,56,55)(32,59,46,58)(40,62,47,61)(52,64,60,63)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=h,d^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289670);