Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451047 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15984 |
| 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 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,17,9,18,41,29,33)(4,10,22,36,23,38,21,37)(5,24,26,40,27,11,30,53)(15,34,44,57,45,42,19,31)(16,39,55,58,46,51,25,32)(20,35,48,61,49,52,50,60)(43,62,47,64,63,59,54,56);(1,3)(2,9)(4,19)(5,25)(6,29)(7,18)(8,17)(10,34)(11,39)(12,41)(13,33)(14,28)(15,22)(16,26)(20,43)(21,45)(23,44)(24,51)(27,55)(30,46)(31,37)(32,40)(35,56)(36,57)(38,42)(47,48)(49,63)(50,54)(52,64)(53,58)(59,60)(61,62);(1,4)(2,10)(3,15)(5,20)(6,21)(7,23)(8,22)(9,31)(11,35)(12,36)(13,38)(14,42)(16,43)(17,44)(18,45)(19,29)(24,52)(25,54)(26,48)(27,49)(28,37)(30,50)(32,56)(33,57)(34,41)(39,62)(40,60)(46,63)(47,55)(51,59)(53,61)(58,64);(1,5,6,26,7,27,8,30)(2,11,12,40,13,24,28,53)(3,16,17,25,18,46,29,55)(4,20,21,48,23,49,22,50)(9,32,14,39,33,58,41,51)(10,35,36,60,38,52,37,61)(15,43,44,54,45,63,19,47)(31,56,42,62,57,64,34,59)Polycyclic presentation:
2^9|a^2=gh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358243);