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 | |
4244 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,25)(19,20,21,24)(22,30,31,26)(27,32,29,28)(33,34,38,42)(35,41,37,39)(36,40,45,48)(43,47,44,46);(3,9)(7,13)(10,15)(14,16)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,32,30,28)(33,35,38,37)(34,39,42,41)(36,43,45,44)(40,46,48,47);(2,6)(7,13)(8,11)(14,16)(18,25)(19,21)(26,30)(27,29)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,23,24)(18,19,25,21)(22,28,31,32)(26,27,30,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,32,30,28)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,31)(18,26,25,30)(19,27,21,29)(20,28,24,32)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002211);