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 | #51146 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,18)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31)(33,34,39,44)(35,45,48,40)(36,38,43,41)(37,42,46,47);(2,6)(7,10)(8,12)(14,16)(17,19,23,21)(18,24,25,27)(20,28,30,29)(22,26,32,31)(33,35,39,48)(34,40,44,45)(36,37,43,46)(38,47,41,42);(2,6)(3,9)(4,13)(14,16)(18,25)(19,21)(20,30)(22,32)(36,43)(37,46)(38,41)(42,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,23,30)(18,26,25,31)(19,29,21,28)(22,27,32,24)(33,36,39,43)(34,41,44,38)(35,46,48,37)(40,47,45,42);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,19)(18,27,25,24)(20,29,30,28)(22,31,32,26)(33,37,39,46)(34,42,44,47)(35,36,48,43)(38,40,41,45);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,22)(18,28)(19,26)(20,27)(21,31)(23,32)(24,30)(25,29)(33,38,39,41)(34,43,44,36)(35,47,48,42)(37,40,46,45)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966412);