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:
(2,5)(3,4)(10,13)(11,12)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,32)(29,31)(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);(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(18,23)(24,27)(25,26)(31,32)(35,44)(36,48)(37,40)(41,47)(46,53)(50,61)(52,55)(57,60);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32)(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);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,19,22,30)(18,24,23,27)(20,28,21,29)(25,31,26,32)(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);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,20,22,21)(18,25,23,26)(19,28,30,29)(24,31,27,32)(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);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,21,22,20)(18,26,23,25)(19,29,30,28)(24,32,27,31)(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=h,c^2=h,d^2=i,e^2=i,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, 8324190);