Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14796 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,26,37)(3,13,16,29,17,35,24,8)(4,9,19,32,20,33,51,59)(5,21,22,36,23,10,25,34)(14,40,42,27,43,38,49,30)(15,39,44,28,45,55,46,52)(18,31,47,57,48,53,50,58)(41,60,62,54,63,61,64,56);(1,3)(2,8)(4,14)(5,15)(6,24)(7,17)(9,30)(10,28)(11,35)(12,29)(13,37)(16,26)(18,41)(19,49)(20,43)(21,52)(22,46)(23,45)(25,44)(27,33)(31,56)(32,38)(34,39)(36,55)(40,59)(42,51)(47,64)(48,63)(50,62)(53,54)(57,61)(58,60);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,27)(10,31)(11,32)(12,33)(13,38)(15,41)(16,42)(17,43)(21,53)(22,47)(23,48)(24,49)(25,50)(26,51)(28,54)(29,30)(34,57)(35,40)(36,58)(37,59)(39,60)(44,62)(45,63)(46,64)(52,56)(55,61);(1,5,6,25,7,23,26,22)(2,10,11,36,12,21,37,34)(3,15,24,44,17,45,16,46)(4,18,19,50,20,48,51,47)(8,28,35,55,29,52,13,39)(9,31,32,58,33,53,59,57)(14,41,49,62,43,63,42,64)(27,54,40,61,30,56,38,60)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,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, 307329);