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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,8,13,15)(10,16,12,14)(17,18,22,25)(19,26,21,23)(20,24,28,31)(27,32,29,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,3,5,11)(2,6,7,9)(4,10,13,12)(8,14,15,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,7)(6,9)(8,15)(14,16)(20,28)(24,31)(27,29)(30,32)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,13)(14,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441928);