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 | |
4020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(19,25)(22,26)(27,31)(30,32)(33,34,38,39)(35,42,37,40)(36,44,45,41)(43,47,46,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18,21,24)(19,26,25,22)(20,23,29,28)(27,32,31,30)(34,39)(36,45)(40,42)(43,46);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,35,38,37)(34,40,39,42)(36,46,45,43)(41,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,26)(28,32)(29,31)(33,36,38,45)(34,41,39,44)(35,43,37,46)(40,47,42,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37,38,35)(34,40,39,42)(36,46,45,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8538123);