Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396834 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14739 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,33)(4,21,25,40)(5,26,29,11)(6,13,7,12)(9,19,34,18)(10,30,37,24)(16,35,22,31)(17,45,48,42)(23,53,55,51)(27,38,28,39)(32,46,56,47)(36,54,57,58)(41,49,50,44)(43,59,52,60)(61,62,63,64);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,24,49,25,16,30,44)(5,17,28,47,29,48,27,46)(10,35,21,50,37,31,40,41)(11,32,39,45,26,56,38,42)(23,52,54,64,55,43,58,62)(36,60,51,63,57,59,53,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31)(11,36)(12,21)(13,40)(14,37)(15,41)(17,43)(18,44)(19,49)(20,22)(26,57)(27,54)(28,58)(29,55)(32,59)(33,50)(34,35)(38,51)(39,53)(42,61)(45,63)(46,62)(47,64)(48,52)(56,60);(1,5,6,27,8,29,7,28)(2,11,12,38,14,26,13,39)(3,17,18,46,20,48,19,47)(4,23,24,54,25,55,30,58)(9,32,15,42,34,56,33,45)(10,36,21,51,37,57,40,53)(16,43,44,62,22,52,49,64)(31,59,41,61,35,60,50,63)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304031);