Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207866 |
| 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 | |
1065 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,24)(19,22,25,30)(20,27,29,23)(26,32,28,31)(33,34)(35,44)(36,50)(37,41)(38,43)(39,49)(40,51)(42,46)(45,59)(47,62)(48,55)(52,61)(53,60)(54,58)(56,64)(57,63);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19)(18,22)(20,28)(21,25)(23,32)(24,30)(26,29)(27,31)(35,45)(36,51)(39,55)(40,50)(44,59)(48,49)(53,54)(58,60);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(19,25)(22,30)(26,28)(31,32)(33,35,46,48,38,45,37,49)(34,39,41,59,43,55,42,44)(36,52,53,63,51,47,54,64)(40,57,60,61,50,56,58,62);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(17,20,21,29)(18,23,24,27)(19,26,25,28)(22,31,30,32)(33,36,38,51)(34,40,43,50)(35,47,45,52)(37,54,46,53)(39,56,55,57)(41,60,42,58)(44,61,59,62)(48,64,49,63);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,25)(22,30)(26,28)(31,32)(33,37,38,46)(34,41,43,42)(35,48,45,49)(36,53,51,54)(39,44,55,59)(40,58,50,60)(47,63,52,64)(56,62,57,61)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=fgi,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^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7137326);