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 | |
4032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,27)(20,25)(21,31)(22,24)(23,28)(26,29)(30,32)(33,34,38,36)(35,39,43,40)(37,44,45,41)(42,47,48,46);(4,11)(8,14)(10,15)(12,16)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(2,7)(3,9)(8,14)(10,15)(18,24)(23,27)(25,32)(26,31)(33,35,38,43)(34,40,36,39)(37,42,45,48)(41,47,44,46);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,30)(18,25,24,32)(19,21,28,29)(23,26,27,31)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,32)(28,30)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,e^2=gi,f^2=i,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, 9627632);