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 | |
4180 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)(7,11)(10,16)(12,14)(13,15)(17,18,22,25)(19,23,28,24)(20,21,30,32)(26,27,29,31)(34,39)(36,40)(41,45)(44,47);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,19)(18,23)(20,29)(21,31)(22,28)(24,25)(26,30)(27,32)(33,34,38,39)(35,40,43,36)(37,45,46,41)(42,44,48,47);(2,4)(5,14)(7,9)(10,13)(17,19,22,28)(18,24,25,23)(20,29,30,26)(21,27,32,31)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,47,45,44);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,30)(31,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,20)(18,21)(19,26)(22,30)(23,27)(24,31)(25,32)(28,29)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,21)(18,20)(19,27)(22,32)(23,26)(24,29)(25,30)(28,31)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669143);