Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15377 |
| 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 | |
1134 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)(4,10)(5,26)(6,13)(7,12)(8,14)(9,18)(11,28)(16,43)(17,32)(19,34)(20,33)(21,31)(22,51)(23,42)(24,37)(25,38)(27,44)(29,41)(30,40)(35,46)(36,54)(39,48)(45,61)(47,55)(49,59)(50,63)(52,62)(53,56)(57,60)(58,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,21,23,46,25,49,24,16)(5,27,28,47,29,48,30,17)(10,35,37,31,38,43,42,59)(11,39,26,55,40,44,41,32)(22,45,52,57,53,64,54,50)(36,61,56,63,62,58,51,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,42)(14,38)(15,43)(17,45)(18,21)(19,49)(20,46)(26,56)(27,57)(28,52)(29,53)(30,54)(32,58)(33,35)(34,59)(39,60)(40,62)(41,51)(44,63)(47,64)(48,50)(55,61);(1,5,6,30,8,29,7,28)(2,11,12,41,14,40,13,26)(3,17,18,48,20,47,19,27)(4,22,23,54,25,53,24,52)(9,32,15,44,34,55,33,39)(10,36,37,51,38,62,42,56)(16,45,21,50,46,64,49,57)(31,58,43,63,59,61,35,60)Polycyclic presentation:
2^9|b^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335277);