Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441222 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15674 |
| 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 | |
1182 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)(3,14,18,9)(4,19,24,38)(5,25,28,41)(6,13)(7,30)(8,12)(10,22,37,23)(11,27,40,26)(15,43)(16,47)(17,34,29,33)(20,42)(21,36)(31,45)(32,49)(35,46)(39,48)(44,62,63,57)(50,56)(51,54)(52,58,64,59)(53,61)(55,60);(1,3,7,17,8,18,6,29)(2,9,30,33,12,14,13,34)(4,20,22,46,24,45,23,15)(5,16,26,49,28,50,27,48)(10,35,19,31,37,43,38,42)(11,39,41,47,40,32,25,56)(21,52,54,63,55,64,53,44)(36,59,51,62,60,58,61,57);(1,4)(2,10,12,37)(3,15)(5,21)(6,22)(7,23)(8,24)(9,31,14,42)(11,36,40,60)(13,19,30,38)(16,44)(17,45)(18,46)(20,29)(25,51,41,61)(26,53)(27,54)(28,55)(32,57,39,62)(33,35,34,43)(47,59,56,58)(48,52)(49,64)(50,63);(1,5,8,28)(2,11)(3,16,18,50)(4,21,24,55)(6,27,7,26)(9,32)(10,36)(12,40)(13,25)(14,39)(15,44,46,63)(17,49,29,48)(19,51)(20,52,45,64)(22,54,23,53)(30,41)(31,57)(33,56)(34,47)(35,58)(37,60)(38,61)(42,62)(43,59)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,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, 348419);