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 | |
4255 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,7)(6,13)(8,11)(14,15)(18,20)(19,26)(25,28)(27,32)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065690);