Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(19,25)(20,27)(22,28)(23,30)(33,34)(35,43)(36,46)(37,49)(38,52)(39,40)(41,50)(42,53)(44,47)(45,48)(51,54)(55,57)(56,59)(58,63)(60,64)(61,62);(2,6)(3,10)(8,12)(11,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(34,40)(36,43)(37,50)(42,52)(44,57)(48,56)(51,62)(58,64);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35,39,46)(34,36,40,43)(37,44,50,57)(38,45,53,59)(41,47,49,55)(42,48,52,56)(51,58,62,64)(54,60,61,63);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36)(34,35)(37,47)(38,48)(39,43)(40,46)(41,44)(42,45)(49,57)(50,55)(51,60)(52,59)(53,56)(54,58)(61,64)(62,63);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37)(34,41)(35,44)(36,47)(38,51)(39,50)(40,49)(42,54)(43,55)(45,58)(46,57)(48,60)(52,61)(53,62)(56,63)(59,64);(33,38,39,53)(34,42,40,52)(35,45,46,59)(36,48,43,56)(37,51,50,62)(41,54,49,61)(44,58,57,64)(47,60,55,63)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022652);