Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441632 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15688 |
| 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 | |
860 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)(3,13,17,41)(4,9,21,34)(5,22,25,10)(6,11)(7,12)(8,16,31,49)(14,42)(15,40)(18,28)(19,32)(20,33,26,37)(23,30)(24,38,27,36)(29,47)(35,46)(39,45)(43,59,63,62)(44,57)(48,58)(50,64,55,56)(51,60)(52,54)(53,61);(1,3,6,16,7,17,18,49)(2,8,28,41,12,31,11,13)(4,14,20,44,21,45,26,47)(5,23,24,46,25,48,27,15)(9,29,37,39,34,57,33,42)(10,35,38,30,22,40,36,58)(19,50,51,63,52,55,53,43)(32,59,61,56,54,62,60,64);(1,4,18,20,7,21,6,26)(2,9,11,37,12,34,28,33)(3,14,16,47,17,45,49,44)(5,19,24,53,25,52,27,51)(8,29,41,42,31,57,13,39)(10,32,38,60,22,54,36,61)(15,43,48,50,46,63,23,55)(30,56,35,62,58,64,40,59);(1,5)(2,10)(3,15)(4,19)(6,27)(7,25)(8,30)(9,32)(11,38)(12,22)(13,40)(14,43)(16,48)(17,46)(18,24)(20,53)(21,52)(23,49)(26,51)(28,36)(29,56)(31,58)(33,61)(34,54)(35,41)(37,60)(39,59)(42,62)(44,55)(45,63)(47,50)(57,64)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348829);