Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #450134 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15958 |
| 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 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,34,19,35)(4,21,25,39,26,38,24,10)(5,27,29,41,31,11,30,42)(16,43,22,36,45,32,48,60)(17,40,46,59,47,56,28,33)(23,51,53,57,55,49,54,37)(44,62,63,64,52,58,50,61);(1,3)(2,9)(4,22,26,48)(5,28)(6,19)(7,18)(8,20)(10,36,39,60)(11,40)(12,34)(13,15)(14,35)(16,25,45,24)(17,29)(21,32,38,43)(23,52,55,44)(27,56)(30,47)(31,46)(33,41)(37,62,57,58)(42,59)(49,61,51,64)(50,54,63,53);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,37)(12,21)(13,38)(14,39)(15,36)(17,44)(18,22)(19,48)(20,45)(27,57)(28,50)(29,53)(30,54)(31,55)(33,58)(34,60)(35,43)(40,61)(41,49)(42,51)(46,63)(47,52)(56,64)(59,62);(1,5,7,30,8,31,6,29)(2,11,13,42,14,27,12,41)(3,17,19,46,20,47,18,28)(4,23,25,54,26,55,24,53)(9,33,34,56,35,59,15,40)(10,37,38,51,39,57,21,49)(16,44,48,63,45,52,22,50)(32,58,60,64,43,62,36,61)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357331);