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 | |
4090 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18)(19,25)(20,24)(21,26)(22,27)(23,30)(28,29)(31,32)(33,34)(35,43)(36,40)(37,47)(38,42)(39,44)(41,45)(46,48);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,47,48,41);(2,8)(3,4)(9,14)(10,12)(33,34,38,42)(35,39,44,43)(36,40,46,48)(37,41,45,47);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20,22,26)(18,24,27,21)(19,28,30,32)(23,31,25,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,44)(45,47)(46,48);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,19)(18,25)(20,28)(21,31)(22,30)(23,27)(24,29)(26,32)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,41,43,47);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,21,22,24)(18,26,27,20)(19,29,30,31)(23,32,25,28)(33,37)(34,41)(35,36)(38,45)(39,40)(42,47)(43,48)(44,46)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=gh,e^2=hi,f^2=gh,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, 9633591);