Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15468 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,9,17,33,16,34,29,32)(4,18,22,37,23,48,21,10)(5,11,25,40,26,38,28,39)(14,41,19,30,43,35,50,58)(15,44,46,60,24,31,45,59)(20,51,53,36,54,47,55,62)(42,63,49,57,56,61,52,64);(1,3,6,17,8,16,7,29)(2,9,13,32,12,34,27,33)(4,19,21,43,23,50,22,14)(5,24,25,46,26,15,28,45)(10,35,18,30,37,41,48,58)(11,31,39,59,38,44,40,60)(20,52,53,56,54,49,55,42)(36,61,51,64,62,63,47,57);(1,4)(2,10,12,37)(3,14,16,43)(5,20)(6,21)(7,22)(8,23)(9,30)(11,36,38,62)(13,18,27,48)(15,42,24,56)(17,19,29,50)(25,53)(26,54)(28,55)(31,57)(32,41)(33,35)(34,58)(39,51,40,47)(44,64)(45,52,46,49)(59,61)(60,63);(1,5)(2,11)(3,15)(4,20)(6,28)(7,25)(8,26)(9,31)(10,36)(12,38)(13,40)(14,42)(16,24)(17,46)(18,47)(19,49)(21,55)(22,53)(23,54)(27,39)(29,45)(30,57)(32,60)(33,59)(34,44)(35,61)(37,62)(41,63)(43,56)(48,51)(50,52)(58,64)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 340091);