Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423621 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15315 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1508 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)(4,9)(5,22)(6,26)(7,12)(8,16)(10,25)(11,28)(14,39)(15,35)(17,37)(18,38)(19,45)(20,49)(21,34)(23,30)(24,36)(27,31)(29,44)(32,48)(33,50)(40,61)(41,57)(42,51)(43,59)(46,56)(47,60)(52,55)(53,64)(54,58)(62,63);(1,3)(2,8)(4,18)(5,23)(6,27)(7,17)(9,29)(10,35)(11,37)(12,31)(13,26)(14,21)(15,24)(16,28)(19,46)(20,41)(22,51)(25,54)(30,36)(32,59)(33,39)(34,57)(38,49)(40,62)(42,55)(43,47)(44,50)(45,61)(48,53)(52,58)(56,60)(63,64);(1,4,7,21)(2,9,12,34)(3,14)(5,19,25,48)(6,20,28,50)(8,29)(10,32,22,45)(11,33,26,49)(13,38)(15,40)(16,41)(17,18)(23,53)(24,47,55,62)(27,44)(30,56)(31,57)(35,59)(36,60,52,63)(37,39)(42,43)(46,54)(51,61)(58,64);(1,5,6,24,7,25,28,55)(2,10,11,36,12,22,26,52)(3,15,16,23,17,42,27,54)(4,19,20,47,21,48,50,62)(8,30,13,35,31,58,37,51)(9,32,33,60,34,45,49,63)(14,40,41,53,18,43,44,46)(29,56,38,59,57,64,39,61)Polycyclic presentation:
2^9|c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330818);