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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34,38,39)(35,36,43,40)(37,41,46,47)(42,44,48,45);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(34,39)(36,40)(41,47)(44,45);(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,45)(44,47)(46,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,36,38,40)(34,35,39,43)(37,45,46,44)(41,48,47,42);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,37,38,46)(34,41,39,47)(35,42,43,48)(36,44,40,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343941);