Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460565 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16367 |
| 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 | |
891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,28,38)(3,8,16,32,15,31,25,39)(4,17,20,40,21,9,26,35)(5,22,23,41,24,10,27,37)(13,42,18,51,44,29,49,33)(14,45,48,58,47,30,50,57)(19,36,53,61,54,34,55,60)(43,63,64,59,46,56,52,62);(1,3)(2,8,11,31)(4,18)(5,14)(6,25)(7,15)(9,33,17,51)(10,30,22,45)(12,39,38,32)(13,20)(16,28)(19,52)(21,49)(23,50)(24,47)(26,44)(27,48)(29,40,42,35)(34,59,36,62)(37,58,41,57)(43,55)(46,53)(54,64)(56,60,63,61);(1,4)(2,9)(3,13)(5,19)(6,26)(7,21)(8,29)(10,36)(11,17)(12,40)(14,46)(15,44)(16,49)(18,25)(20,28)(22,34)(23,55)(24,54)(27,53)(30,56)(31,42)(32,51)(33,39)(35,38)(37,60)(41,61)(43,47)(45,63)(48,64)(50,52)(57,59)(58,62);(1,5,7,24)(2,10,11,22)(3,14,15,47)(4,19,21,54)(6,27,28,23)(8,30,31,45)(9,34,17,36)(12,41,38,37)(13,43,44,46)(16,50,25,48)(18,52,49,64)(20,55,26,53)(29,56,42,63)(32,58,39,57)(33,59,51,62)(35,61,40,60)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367762);