Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405494 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14913 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,38)(4,18,22,9)(5,10,24,35)(6,25,26,11)(8,27,30,16)(14,39,19,36)(15,37,43,59)(20,32,50,44)(21,33,46,45)(23,51,52,34)(28,41,31,48)(29,53,55,42)(40,58,47,60)(49,62,63,57)(54,64,56,61);(1,3,6,16,7,17,26,27)(2,8,11,13,12,30,25,38)(4,19,46,41,22,14,21,48)(5,15,23,42,24,43,52,53)(9,31,45,36,18,28,33,39)(10,29,34,37,35,55,51,59)(20,47,63,61,50,40,49,64)(32,56,62,58,44,54,57,60);(1,4,6,21,7,22,26,46)(2,9,11,33,12,18,25,45)(3,14,16,41,17,19,27,48)(5,20,23,49,24,50,52,63)(8,28,13,36,30,31,38,39)(10,32,34,57,35,44,51,62)(15,40,42,61,43,47,53,64)(29,54,37,58,55,56,59,60);(1,5)(2,10)(3,15)(4,20)(6,23)(7,24)(8,29)(9,32)(11,34)(12,35)(13,37)(14,40)(16,42)(17,43)(18,44)(19,47)(21,49)(22,50)(25,51)(26,52)(27,53)(28,54)(30,55)(31,56)(33,57)(36,58)(38,59)(39,60)(41,61)(45,62)(46,63)(48,64)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312691);