Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16297 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1511 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)(3,13,16,8)(4,9)(5,22,24,37)(6,12)(7,11)(10,25,35,26)(14,28,18,31)(15,42)(17,39,46,30)(19,49,23,54)(20,33)(21,34)(27,36)(29,44)(32,50,59,52)(38,47)(40,61)(41,56,48,58)(43,64)(45,53)(51,60)(55,63)(57,62);(1,3)(2,8)(4,18)(5,15)(6,17)(7,16)(9,31)(10,29)(11,13)(12,30)(14,20)(19,43)(21,48)(22,53)(23,40)(24,45)(25,44)(26,47)(27,46)(28,33)(32,57)(34,58)(35,38)(36,39)(37,42)(41,51)(49,64)(50,63)(52,62)(54,61)(55,59)(56,60);(1,4)(2,9)(3,14,16,18)(5,23,24,19)(6,21)(7,20)(8,28,13,31)(10,32,35,59)(11,33)(12,34)(15,43)(17,41,46,48)(22,49,37,54)(25,52,26,50)(27,51)(29,55)(30,56,39,58)(36,60)(38,57)(40,45)(42,64)(44,63)(47,62)(53,61);(1,5,7,24)(2,10,11,35)(3,15,16,45)(4,19,20,23)(6,26,27,25)(8,29,13,38)(9,32,33,59)(12,37,36,22)(14,40,18,43)(17,47,46,44)(21,52,51,50)(28,55,31,57)(30,42,39,53)(34,54,60,49)(41,63,48,62)(56,64,58,61)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364786);