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 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,22,23)(19,27,28,24)(20,30,31,25)(21,29,32,26)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(2,4)(5,13)(7,9)(11,16)(18,23)(21,32)(24,27)(25,30)(34,40)(35,44)(37,47)(42,45);(3,10)(7,9)(11,16)(14,15)(18,23)(19,28)(25,30)(26,29)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,19,22,28)(18,24,23,27)(20,29,31,26)(21,25,32,30)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,32)(18,26,23,29)(19,25,28,30)(20,27,31,24)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=hi,e^2=gh,f^2=g,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, 8662429);