Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504434 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26601 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13340 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1773 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,14,9,16)(3,17,20,40,23,39,22,10)(4,12,5,27,25,11,28,42)(8,32,30,48,31,15,35,45)(18,49,52,63,53,64,58,36)(19,37,24,59,55,41,56,50)(21,46,57,38,54,51,33,62)(26,43,29,47,60,44,34,61);(1,3)(2,10)(4,24)(5,19)(6,22)(7,23)(8,33)(9,20)(11,41)(12,37)(13,39)(14,40)(15,46)(16,17)(18,26)(21,35)(25,56)(27,50)(28,55)(29,58)(30,54)(31,57)(32,51)(34,52)(36,43)(38,48)(42,59)(44,63)(45,62)(47,64)(49,61)(53,60);(1,4)(2,11)(3,18)(5,6)(7,25)(8,29)(9,28)(10,36)(12,14)(13,42)(15,47)(16,27)(17,49)(19,54)(20,52)(21,55)(22,58)(23,53)(24,33)(26,35)(30,60)(31,34)(32,61)(37,51)(38,50)(39,64)(40,63)(41,46)(43,48)(44,45)(56,57)(59,62);(1,5,6,4,7,28,9,25)(2,12,13,42,14,11,16,27)(3,19,22,24,23,55,20,56)(8,34,30,60,31,29,35,26)(10,37,39,59,40,41,17,50)(15,43,45,61,32,44,48,47)(18,33,52,54,53,57,58,21)(36,62,49,51,63,38,64,46)Polycyclic presentation:
2^9|a^2=e,d^2=fh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,e^d=ei,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411630);