Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2712 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15784 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,29)(3,14,17,33,18,9,30,34)(4,19,23,37,24,46,22,10)(5,25,27,54,28,11,26,39)(15,35,20,40,42,57,48,31)(16,38,43,58,44,32,55,59)(21,49,51,36,52,45,50,53)(41,60,47,62,63,61,64,56);(1,3)(2,9)(4,20)(5,16)(6,17)(7,30)(8,18)(10,35)(11,38)(12,14)(13,33)(15,23)(19,31)(21,47)(22,42)(24,48)(25,32)(26,43)(27,55)(28,44)(29,34)(36,61)(37,57)(39,59)(40,46)(41,51)(45,62)(49,56)(50,63)(52,64)(53,60)(54,58);(1,4,7,23,8,24,6,22)(2,10,29,46,12,37,13,19)(3,15,17,20,18,42,30,48)(5,21,27,51,28,52,26,50)(9,31,33,57,14,40,34,35)(11,36,54,49,25,53,39,45)(16,41,43,47,44,63,55,64)(32,56,58,61,38,62,59,60);(1,5)(2,11,12,25)(3,16)(4,21)(6,26)(7,27)(8,28)(9,32,14,38)(10,36,37,53)(13,39,29,54)(15,41)(17,43)(18,44)(19,45,46,49)(20,47)(22,50)(23,51)(24,52)(30,55)(31,56,40,62)(33,58,34,59)(35,60,57,61)(42,63)(48,64)Polycyclic presentation:
2^9|a^2=g,c^2=eg,d^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,g^b=gi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351846);