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 | |
3645 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)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(35,43)(39,44)(45,48)(46,57)(53,55)(54,58)(59,61)(63,64);(2,7)(4,9)(8,13)(12,15)(21,29)(25,28)(27,30)(31,32)(33,34,38,42)(35,39,43,44)(36,47,50,40)(37,41,52,51)(45,55,48,53)(46,54,57,58)(49,60,62,56)(59,64,61,63);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,34)(35,44)(36,40)(37,51)(38,42)(39,43)(41,52)(45,55)(46,54)(47,50)(48,53)(49,60)(56,62)(57,58)(59,63)(61,64);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,35)(34,39)(36,48)(37,46)(38,43)(40,55)(41,54)(42,44)(45,50)(47,53)(49,61)(51,58)(52,57)(56,64)(59,62)(60,63);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,36,38,50)(34,40,42,47)(35,45,43,48)(37,49,52,62)(39,53,44,55)(41,56,51,60)(46,59,57,61)(54,63,58,64);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,37)(34,41)(35,46)(36,49)(38,52)(39,54)(40,56)(42,51)(43,57)(44,58)(45,59)(47,60)(48,61)(50,62)(53,63)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839404);