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 | |
4135 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)(4,6)(5,8)(7,10)(11,15)(12,13)(14,16)(19,25)(22,26)(27,31)(30,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,4)(5,13)(7,9)(11,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(3,10)(7,9)(11,16)(14,15)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664242);