Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232546 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #2815 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8)(3,14)(4,19)(5,23)(9,32)(10,36)(11,20)(12,28)(13,31)(15,45)(17,51)(18,44)(21,27)(22,54)(24,25)(29,37)(30,35)(33,58)(34,40)(38,60)(39,53)(41,47)(43,56)(46,61)(48,59)(50,64)(52,55)(62,63);(1,2,7,13)(3,15,18,10)(4,20,21,53)(5,9,25,35)(6,11,26,39)(8,27,31,19)(12,23,40,24)(14,41,44,37)(16,45,49,36)(17,46,51,33)(22,30,54,32)(28,52,34,55)(29,42,47,57)(38,43,48,56)(50,60,63,59)(58,64,61,62);(2,9)(4,21)(5,24)(8,28)(10,33)(11,12)(13,35)(15,46)(16,49)(17,43)(19,27)(20,32)(22,55)(23,25)(29,58)(30,53)(31,34)(36,60)(37,38)(39,40)(41,48)(42,57)(45,59)(47,61)(50,64)(51,56)(52,54)(62,63);(1,3,7,18)(2,10,13,15)(4,16,21,49)(5,17,25,51)(6,14,26,44)(8,29,31,47)(9,33,35,46)(11,37,39,41)(12,38,40,48)(19,42,27,57)(20,36,53,45)(22,50,54,63)(23,56,24,43)(28,58,34,61)(30,59,32,60)(52,62,55,64)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139751);