Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455254 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16261 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,27,7,12,29,11)(3,13,16,8,17,41,50,32)(4,18,21,34,22,9,28,38)(5,23,25,37,26,10,35,57)(14,42,44,30,19,39,49,51)(15,45,47,55,48,40,62,31)(20,52,24,36,54,33,59,56)(43,61,46,60,53,63,58,64);(1,3,6,16,7,17,29,50)(2,8,11,13,12,32,27,41)(4,19,21,49,22,14,28,44)(5,15,25,47,26,48,35,62)(9,30,34,42,18,51,38,39)(10,31,37,40,23,55,57,45)(20,53,24,58,54,43,59,46)(33,60,36,61,52,64,56,63);(1,4)(2,9)(3,14)(5,24)(6,28)(7,22)(8,30)(10,36)(11,38)(12,18)(13,39)(15,46)(16,49)(17,19)(20,25)(21,29)(23,56)(26,59)(27,34)(31,61)(32,51)(33,37)(35,54)(40,60)(41,42)(43,47)(44,50)(45,64)(48,58)(52,57)(53,62)(55,63);(1,5,6,25,7,26,29,35)(2,10,11,37,12,23,27,57)(3,15,16,47,17,48,50,62)(4,20,21,24,22,54,28,59)(8,31,13,40,32,55,41,45)(9,33,34,36,18,52,38,56)(14,43,44,46,19,53,49,58)(30,60,42,61,51,64,39,63)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362450);