Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418531 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1638 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)(3,15)(4,21)(5,26)(6,14)(7,12)(8,13)(9,18)(10,24)(11,29)(16,31)(17,32)(19,33)(20,34)(22,36)(23,38)(25,37)(27,56)(28,41)(30,40)(35,46)(39,47)(42,45)(43,62)(44,49)(48,55)(50,52)(51,57)(53,61)(54,60)(58,64)(59,63);(1,3,8,20)(2,9,13,33)(4,16,25,46)(5,27,30,47)(6,19,7,18)(10,35,38,31)(11,39,41,56)(12,34,14,15)(17,29,48,28)(21,49,37,42)(22,51,53,64)(23,45,24,44)(26,55,40,32)(36,59,61,62)(43,54,63,52)(50,57,60,58);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,38)(14,21)(15,42)(17,43)(18,44)(19,45)(20,46)(26,50)(27,51)(28,52)(29,54)(30,53)(32,57)(33,35)(34,49)(39,62)(40,60)(41,61)(47,64)(48,63)(55,58)(56,59);(1,5,6,28,8,30,7,29)(2,11,12,40,13,41,14,26)(3,17,18,27,20,48,19,47)(4,22,23,52,25,53,24,54)(9,32,15,39,33,55,34,56)(10,36,37,60,38,61,21,50)(16,43,44,51,46,63,45,64)(31,57,42,62,35,58,49,59)Polycyclic presentation:
2^9|b^2=i,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325728);