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 | |
4130 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)(19,25)(20,24)(21,26)(22,23)(27,29)(28,30)(31,32)(33,34,38,39)(35,40,43,36)(37,45,47,41)(42,44,48,46);(3,9)(7,13)(10,15)(14,16)(17,18)(19,21)(20,28)(22,23)(24,30)(25,26)(27,29)(31,32)(34,39)(36,40)(41,45)(44,46);(2,6)(7,13)(8,11)(14,16)(18,23)(19,26)(24,28)(27,32)(33,35,38,43)(34,36,39,40)(37,42,47,48)(41,44,45,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,31)(28,29)(30,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,46)(44,45)(47,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,28)(25,29)(26,32)(33,36,38,40)(34,35,39,43)(37,46,47,44)(41,48,45,42);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,29)(22,25)(23,26)(24,32)(27,28)(30,31)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9621985);