Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,16)(3,17,21,36)(4,22,25,54)(5,26,30,11)(6,13,32,53)(8,15,18,14)(9,34,40,19)(10,31,45,24)(12,47,46,28)(20,44,57,43)(23,39,27,38)(29,50,63,35)(33,52,55,51)(37,61,48,56)(41,62,64,60)(42,59,49,58);(1,3,18,25)(2,9,14,38)(4,7,21,8)(5,23,47,34)(6,31,55,20)(10,41,43,35)(11,36,12,22)(13,49,51,37)(15,39,16,40)(17,46,54,26)(19,30,27,28)(24,33,57,32)(29,61,62,58)(42,52,48,53)(44,50,45,64)(56,60,59,63);(1,4,8,3)(2,10,15,44)(5,27,28,34)(6,24,33,20)(7,25,18,21)(9,35,39,64)(11,42,46,37)(12,48,26,49)(13,22,52,17)(14,43,16,45)(19,30,23,47)(29,59,60,61)(31,55,57,32)(36,53,54,51)(38,41,40,50)(56,63,58,62);(2,11)(5,28)(6,32)(9,36)(10,42)(12,14)(13,50)(15,46)(16,26)(17,40)(19,23)(20,57)(22,38)(24,31)(27,34)(29,62)(30,47)(33,55)(35,53)(37,44)(39,54)(41,52)(43,48)(45,49)(51,64)(56,59)(58,61)(60,63)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153582);