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 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,14,12)(10,15,13,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,39,42)(35,40,46,41)(36,47,37,38)(43,45,44,48);(1,3)(2,7)(4,11)(5,13)(6,9)(8,16)(10,14)(12,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,35,39,46)(34,40,42,41)(36,44,37,43)(38,45,47,48);(3,9)(7,11)(10,13)(15,16)(18,21)(23,25)(24,28)(31,32)(33,36,39,37)(34,38,42,47)(35,43,46,44)(40,48,41,45);(1,4,6,2)(3,7,9,11)(5,12,14,8)(10,15,13,16)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,37,39,36)(34,38,42,47)(35,44,46,43)(40,48,41,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,16)(13,14)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,38)(34,36)(35,45)(37,42)(39,47)(40,44)(41,43)(46,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,35)(34,41)(36,43)(37,44)(38,45)(39,46)(40,42)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259640);