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 | |
4196 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(18,23)(19,26)(25,30)(27,32)(33,34,38,39)(35,43,46,40)(36,41,47,45)(37,44,48,42);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28)(34,39)(36,47)(40,43)(42,44);(2,6)(3,4)(5,13)(15,16)(19,26)(20,24)(21,29)(25,30)(33,35,38,46)(34,40,39,43)(36,42,47,44)(37,41,48,45);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,35)(34,40)(36,42)(37,45)(38,46)(39,43)(41,48)(44,47);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,36)(34,41)(35,44)(37,40)(38,47)(39,45)(42,46)(43,48);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,44)(41,46)(43,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8934990);