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 | #57755 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,5)(6,8)(9,10,15,13)(11,14,18,16)(12,20,22,17)(19,23,24,21)(28,39)(33,42)(36,49)(40,52)(44,47)(46,53)(50,56)(54,55);(3,7)(6,8)(10,13)(11,18)(17,20)(19,24)(26,31)(28,39)(29,41)(32,35)(36,49)(37,51)(46,53)(54,55);(1,2)(3,6)(4,5)(7,8)(9,11,15,18)(10,16,13,14)(12,19,22,24)(17,23,20,21)(25,26,30,31)(27,35,38,32)(28,33,39,42)(29,34,41,43)(36,47,49,44)(37,48,51,45)(40,46,52,53)(50,55,56,54);(9,12,15,22)(10,17,13,20)(11,19,18,24)(14,21,16,23)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,41,51)(33,44,42,47)(34,45,43,48)(40,50,52,56)(46,54,53,55);(1,3)(2,6)(4,7)(5,8)(9,13,15,10)(11,14,18,16)(12,20,22,17)(19,21,24,23)(25,28)(26,33)(27,36)(29,40)(30,39)(31,42)(32,44)(34,46)(35,47)(37,50)(38,49)(41,52)(43,53)(45,54)(48,55)(51,56);(1,2)(3,6)(4,5)(7,8)(9,14)(10,11)(12,21)(13,18)(15,16)(17,19)(20,24)(22,23)(25,29)(26,34)(27,37)(28,40)(30,41)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(39,52)(42,53)(44,54)(47,55)(49,56)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662028);