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 | |
4280 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,38,39)(35,40,43,37)(36,44,47,41)(42,46,45,48);(2,7)(4,9)(5,14)(11,16)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(34,39)(37,40)(41,44)(46,48);(2,7)(3,10)(8,13)(11,16)(19,27)(20,29)(23,28)(24,32)(33,35,38,43)(34,40,39,37)(36,42,47,45)(41,48,44,46);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,35)(34,37)(36,45)(38,43)(39,40)(41,48)(42,47)(44,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,37)(34,35)(36,46)(38,40)(39,43)(41,42)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746342);