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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,6,7)(3,9,11,4)(5,13,14,8)(10,12,16,15)(19,21)(23,25)(27,29)(31,32)(33,34,39,43)(35,37,41,50)(36,46,49,40)(38,42,54,58)(44,47,56,51)(45,52,53,57)(48,60,63,55)(59,61,62,64);(2,7)(4,9)(8,13)(12,15)(17,18,22,26)(19,23,21,25)(20,28,30,24)(27,32,29,31)(35,41)(37,50)(44,56)(45,53)(47,51)(52,57)(59,62)(61,64);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32)(33,35)(34,37)(36,44)(38,53)(39,41)(40,51)(42,57)(43,50)(45,54)(46,47)(48,62)(49,56)(52,58)(55,61)(59,63)(60,64);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(33,36,39,49)(34,40,43,46)(35,44,41,56)(37,51,50,47)(38,48,54,63)(42,55,58,60)(45,59,53,62)(52,64,57,61);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,20,22,30)(18,24,26,28)(19,27,21,29)(23,31,25,32)(33,37)(34,41)(35,43)(36,47)(38,52)(39,50)(40,44)(42,53)(45,58)(46,56)(48,61)(49,51)(54,57)(55,59)(60,62)(63,64);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)(33,38)(34,42)(35,45)(36,48)(37,52)(39,54)(40,55)(41,53)(43,58)(44,59)(46,60)(47,61)(49,63)(50,57)(51,64)(56,62)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8946365);