Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2428 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,11,7,13,28,12)(3,14,18,32,19,31,17,8)(4,20,23,9,24,35,53,36)(5,10,26,38,27,40,55,39)(15,33,45,48,21,29,44,41)(16,42,47,37,25,54,46,30)(22,49,51,34,52,58,64,59)(43,57,62,63,50,56,61,60);(1,3)(2,8)(4,21)(5,25)(6,17)(7,19)(9,33)(10,37)(11,31)(12,14)(13,32)(15,24)(16,27)(18,28)(20,48)(22,50)(23,45)(26,47)(29,36)(30,40)(34,57)(35,41)(38,42)(39,54)(43,52)(44,53)(46,55)(49,63)(51,62)(56,59)(58,60)(61,64);(1,4,7,24)(2,9)(3,15)(5,22,27,52)(6,23,28,53)(8,29,32,33)(10,34)(11,35)(12,20)(13,36)(14,41,31,48)(16,43)(17,44)(18,45)(19,21)(25,50)(26,51,55,64)(30,56,37,57)(38,58)(39,49)(40,59)(42,60,54,63)(46,61)(47,62);(1,5,6,26,7,27,28,55)(2,10,11,38,13,40,12,39)(3,16,18,47,19,25,17,46)(4,22,23,51,24,52,53,64)(8,30,14,42,32,37,31,54)(9,34,35,58,36,59,20,49)(15,43,45,62,21,50,44,61)(29,56,41,60,33,57,48,63);(1,6,7,28)(2,11,13,12)(3,17,19,18)(4,23,24,53)(5,26,27,55)(8,31,32,14)(9,35,36,20)(10,38,40,39)(15,44,21,45)(16,46,25,47)(22,51,52,64)(29,48,33,41)(30,54,37,42)(34,58,59,49)(43,61,50,62)(56,63,57,60)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351591);