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 |
| prime radical | #2703 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,13,7)(4,14,10,5)(11,12,16,15)(18,23)(19,25)(26,31)(27,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,6,13)(2,7,8,9)(4,11,10,16)(5,12,14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(35,42)(39,46)(43,47)(44,48);(2,8)(4,10)(7,9)(11,16)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,4,6,10)(2,5,8,14)(3,11,13,16)(7,12,9,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(1,3)(2,9)(4,11)(5,15)(6,13)(7,8)(10,16)(12,14)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5)(2,10)(3,12)(4,8)(6,14)(7,16)(9,11)(13,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9592751);