Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461277 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16385 |
| 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 | |
788 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)(3,15,20,42)(4,21)(5,27)(6,12)(7,13)(8,14)(9,18,34,19)(10,25)(11,29)(16,32,47,52)(17,33,51,61)(22,43,46,35)(23,54)(24,38)(26,37)(28,58)(30,41)(31,40)(36,56)(39,55)(44,50,57,49)(45,60,64,63)(48,62,53,59);(1,3)(2,9,14,34)(4,22)(5,17)(6,18)(7,19)(8,20)(10,35,38,43)(11,33,41,61)(12,15,13,42)(16,24)(21,52,37,32)(23,53)(25,47)(26,46)(27,57,40,44)(28,45)(29,49)(30,50)(31,51)(36,60,54,63)(39,62,58,59)(48,56)(55,64);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,39)(12,37)(13,21)(14,38)(15,43)(17,48)(18,22)(19,46)(20,47)(23,29)(27,54)(30,56)(31,55)(33,60)(34,52)(35,42)(36,40)(41,58)(44,59)(45,49)(50,64)(51,53)(57,62)(61,63);(1,5,6,29,8,31,7,30)(2,11,12,40,14,41,13,27)(3,17,18,49,20,51,19,50)(4,23,24,28,26,56,25,55)(9,33,15,44,34,61,42,57)(10,36,21,39,38,54,37,58)(16,45,46,48,47,64,22,53)(32,59,35,60,52,62,43,63)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368473);