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 | |
4143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(36,44)(40,45)(42,48)(46,47);(1,3)(2,8)(4,5)(6,13)(7,11)(9,16)(10,15)(12,14)(18,19)(23,25)(24,26)(30,31)(34,38)(39,41)(40,45)(46,47);(2,9)(3,11)(4,14)(10,13)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014429);