Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207850 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,10,13,7)(4,8,5,14)(11,16,12,15)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30)(33,34,38,43)(35,44,45,57)(36,48,49,40)(37,41,53,42)(39,54,55,47)(46,62,50,61)(51,60,52,59)(56,64,58,63);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19)(18,22)(20,28)(21,25)(23,31)(24,32)(26,29)(27,30)(35,45)(36,49)(39,55)(40,48)(44,57)(47,54)(51,52)(59,60);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,23)(19,25)(24,27)(26,28)(33,35,37,47,38,45,53,54)(34,39,42,44,43,55,41,57)(36,50,52,63,49,46,51,64)(40,58,59,62,48,56,60,61);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(17,20,21,29)(18,24,23,27)(19,26,25,28)(22,30,31,32)(33,36,38,49)(34,40,43,48)(35,46,45,50)(37,52,53,51)(39,56,55,58)(41,60,42,59)(44,61,57,62)(47,64,54,63);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,25)(22,31)(26,28)(30,32)(33,37,38,53)(34,41,43,42)(35,47,45,54)(36,51,49,52)(39,57,55,44)(40,59,48,60)(46,63,50,64)(56,61,58,62)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fghi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7139123);