Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448911 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15925 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,31)(3,14,17,35,18,34,30,9)(4,19,23,39,24,38,22,10)(5,25,27,56,29,11,28,41)(15,32,44,36,45,59,20,42)(16,46,26,33,47,40,57,60)(21,50,53,37,54,48,52,55)(43,63,64,62,51,58,49,61);(1,3,8,18)(2,9,13,35)(4,20)(5,26,29,57)(6,30,7,17)(10,36)(11,40,25,46)(12,34,31,14)(15,22)(16,28,47,27)(19,32)(21,51)(23,45)(24,44)(33,56,60,41)(37,58)(38,59)(39,42)(43,54)(48,62)(49,52)(50,61)(53,64)(55,63);(1,4,6,22,8,24,7,23)(2,10,12,38,13,39,31,19)(3,15,17,20,18,45,30,44)(5,21,27,52,29,54,28,53)(9,32,14,42,35,59,34,36)(11,37,41,50,25,55,56,48)(16,43,26,49,47,51,57,64)(33,58,40,62,60,63,46,61);(1,5)(2,11,13,25)(3,16,18,47)(4,21)(6,27)(7,28)(8,29)(9,33)(10,37,39,55)(12,41,31,56)(14,40)(15,43,45,51)(17,26,30,57)(19,48,38,50)(20,49,44,64)(22,52)(23,53)(24,54)(32,58)(34,46)(35,60)(36,61)(42,62)(59,63)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356107);