Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13873 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1813 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)(3,13,17,8)(4,18,22,36)(5,23,25,10)(6,12)(7,11)(9,21,33,26)(14,39)(15,29)(16,38,45,30)(19,47)(20,49)(24,53,55,34)(27,35)(28,41)(31,44)(32,50)(37,43)(40,61,48,60)(42,57)(46,51)(52,54)(56,62,58,63)(59,64);(1,3,6,16,7,17,27,45)(2,8,12,30,11,13,35,38)(4,19,21,44,22,14,26,41)(5,15,24,42,25,43,55,64)(9,31,18,39,33,28,36,47)(10,29,34,57,23,37,53,59)(20,48,50,63,51,40,54,62)(32,58,46,60,52,56,49,61);(1,4)(2,9)(3,14)(5,20)(6,26)(7,22)(8,28)(10,32)(11,33)(12,36)(13,31)(15,40)(16,44)(17,19)(18,35)(21,27)(23,52)(24,54)(25,51)(29,56)(30,39)(34,49)(37,58)(38,47)(41,45)(42,63)(43,48)(46,53)(50,55)(57,60)(59,61)(62,64);(1,5,7,25)(2,10,11,23)(3,15,17,43)(4,20,22,51)(6,24,27,55)(8,29,13,37)(9,32,33,52)(12,34,35,53)(14,40,19,48)(16,42,45,64)(18,46,36,49)(21,50,26,54)(28,56,31,58)(30,57,38,59)(39,60,47,61)(41,62,44,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267485);