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 | #2847 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,22,24)(19,25,27,21)(20,28,29,23)(26,30,32,31)(33,34,39,41)(35,40,43,48)(36,38,46,47)(37,45,44,42);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,30,28,31)(33,35,39,43)(34,40,41,48)(36,44,46,37)(38,42,47,45);(2,6)(7,10)(8,12)(14,16)(20,29)(23,28)(26,32)(30,31)(34,41)(35,43)(36,46)(42,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,29)(18,23,24,28)(19,26,27,32)(21,30,25,31)(33,36)(34,38)(35,44)(37,43)(39,46)(40,42)(41,47)(45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,19)(18,21)(20,26)(22,27)(23,30)(24,25)(28,31)(29,32)(33,37,39,44)(34,42,41,45)(35,36,43,46)(38,40,47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,32)(29,31)(33,38)(34,36)(35,45)(37,40)(39,47)(41,46)(42,43)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=h,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, 8659689);