Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16866 |
| 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 | |
787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,13,26,12)(3,14,19,33,17,8,18,42)(4,20,23,37,24,36,29,9)(5,10,30,41,27,39,28,40)(15,45,21,43,47,34,48,31)(16,32,49,59,51,44,50,60)(22,54,56,35,25,52,55,38)(46,62,53,63,64,61,57,58);(1,3,7,17)(2,8)(4,21)(5,16,27,51)(6,19,26,18)(9,34,37,45)(10,32)(11,42)(12,33)(13,14)(15,29)(20,31,36,43)(22,53)(23,47)(24,48)(25,57)(28,50,30,49)(35,61,38,62)(39,44)(40,60)(41,59)(46,55)(52,58,54,63)(56,64);(1,4)(2,9)(3,15)(5,25)(6,29)(7,24)(8,31)(10,38)(11,36)(12,20)(13,37)(14,43)(16,46)(17,47)(18,21)(19,48)(22,27)(23,26)(28,55)(30,56)(32,58)(33,45)(34,42)(35,39)(40,54)(41,52)(44,63)(49,57)(50,53)(51,64)(59,61)(60,62);(1,5,26,28,7,27,6,30)(2,10,12,40,13,39,11,41)(3,16,18,50,17,51,19,49)(4,22,29,55,24,25,23,56)(8,32,33,60,14,44,42,59)(9,35,36,54,37,38,20,52)(15,46,48,57,47,64,21,53)(31,58,34,61,43,63,45,62)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,d^2=efi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380660);