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 | |
4208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,5,11,7)(4,12,13,9)(10,16,14,15)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,48)(45,47);(1,3,6,11)(2,7,8,5)(4,10,13,14)(9,16,12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,48,46,47);(2,8)(4,13)(5,7)(10,14)(19,25)(20,28)(23,27)(24,29)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,48,41,47);(1,4,6,13)(2,9,8,12)(3,10,11,14)(5,15,7,16)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,47,46,48);(1,3)(2,5)(4,14)(6,11)(7,8)(9,16)(10,13)(12,15)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,48,41,47);(1,5)(2,3)(4,15)(6,7)(8,11)(9,10)(12,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,39)(34,35)(36,47)(37,43)(38,48)(40,41)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8829726);