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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3709 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,39,43)(35,40,44,37)(36,47,48,41)(38,42,54,58)(45,51,56,49)(46,55,53,52)(50,60,61,57)(59,64,63,62);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(35,44)(36,48)(37,40)(41,47)(46,53)(50,61)(52,55)(57,60);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35)(34,40)(36,45)(37,43)(38,53)(39,44)(41,49)(42,52)(46,54)(47,51)(48,56)(50,63)(55,58)(57,64)(59,61)(60,62);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,39,48)(34,41,43,47)(35,45,44,56)(37,51,40,49)(38,50,54,61)(42,57,58,60)(46,59,53,63)(52,64,55,62);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(33,37,39,40)(34,35,43,44)(36,49,48,51)(38,52,54,55)(41,56,47,45)(42,46,58,53)(50,62,61,64)(57,63,60,59);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38)(34,42)(35,46)(36,50)(37,52)(39,54)(40,55)(41,57)(43,58)(44,53)(45,59)(47,60)(48,61)(49,62)(51,64)(56,63)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8324183);