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 | |
4197 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,22,24)(19,23,26,20)(21,25,30,31)(27,29,32,28)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(2,7)(3,11)(4,14)(13,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(2,7)(4,14)(8,10)(12,16)(18,24)(19,26)(25,31)(27,32)(37,46)(40,48)(42,45)(44,47);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,20)(18,19)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,18)(19,20)(21,25)(22,24)(23,26)(27,28)(29,32)(30,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,d^2=g,f^2=hi,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, 8540722);