Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13998 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1395 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)(4,9,22,37)(5,24)(6,27)(7,12)(8,16)(10,25)(11,28)(14,40,42,33)(15,43)(17,31)(18,32)(19,48,47,29)(20,50,52,59)(21,36,23,35)(26,54)(30,46)(34,53,60,51)(38,55)(39,44)(41,56,62,63)(45,57)(49,61,64,58);(1,3)(2,8,12,31)(4,19)(5,15)(6,16)(7,18)(9,33,37,40)(10,30,38,43)(11,13,27,32)(14,23)(17,28)(20,49)(21,42)(22,47)(24,39,54,57)(25,44)(26,46)(29,35,48,36)(34,58,60,61)(41,53)(45,55)(50,63,59,56)(51,62)(52,64);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,29)(10,34)(11,35)(12,37)(13,33)(15,41)(16,19)(17,47)(18,42)(23,28)(24,50)(25,51)(26,52)(27,36)(30,56)(31,48)(32,40)(38,60)(39,58)(43,63)(44,49)(45,64)(46,62)(53,55)(54,59)(57,61);(1,5,6,25,7,26,28,55)(2,10,11,24,12,38,27,54)(3,15,16,44,18,46,17,45)(4,20,21,51,22,52,23,53)(8,30,13,39,31,43,32,57)(9,34,35,50,37,60,36,59)(14,41,19,49,42,62,47,64)(29,56,33,58,48,63,40,61)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274289);