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 | |
4174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18,22,25)(19,24,28,23)(20,21,30,31)(26,32,29,27)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,21)(22,25)(24,28)(26,27)(29,32)(30,31)(33,35,39,38)(34,40,43,44)(36,45,42,47)(37,48,41,46);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,30)(31,32)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,46,44,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,25)(26,30)(31,32)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,45,44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,22,30)(18,21,25,31)(19,26,28,29)(23,27,24,32)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,46)(47,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,21)(18,20)(19,27)(22,31)(23,26)(24,29)(25,30)(28,32)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,48,41,46)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411682);