Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14754 |
| 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 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,26,7,12,19,11)(3,13,16,8,17,39,40,29)(4,9,30,33,22,34,21,52)(5,23,24,36,25,10,35,54)(14,37,48,31,18,46,42,27)(15,43,44,53,45,38,59,28)(20,49,56,64,51,32,50,58)(41,61,63,55,47,60,62,57);(1,3)(2,8,12,29)(4,18,22,14)(5,15)(6,16)(7,17)(9,31)(10,28,23,53)(11,13,26,39)(19,40)(20,47,51,41)(21,48,30,42)(24,44)(25,45)(27,34)(32,57)(33,46)(35,59)(36,38,54,43)(37,52)(49,55)(50,63,56,62)(58,61)(60,64);(1,4,19,30,7,22,6,21)(2,9,26,52,12,34,11,33)(3,14,40,48,17,18,16,42)(5,20,35,56,25,51,24,50)(8,27,39,46,29,31,13,37)(10,32,54,64,23,49,36,58)(15,41,59,63,45,47,44,62)(28,55,43,61,53,57,38,60);(1,5,6,24,7,25,19,35)(2,10,11,36,12,23,26,54)(3,15,16,44,17,45,40,59)(4,20,21,50,22,51,30,56)(8,28,13,38,29,53,39,43)(9,32,33,58,34,49,52,64)(14,41,42,62,18,47,48,63)(27,55,37,60,31,57,46,61)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ei,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304943);