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 | |
4042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,8,11,6)(4,12,14,9)(10,15,13,16)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(1,3,5,11)(2,6,7,8)(4,10,14,13)(9,15,12,16)(17,19,23,28)(18,24,25,22)(20,29,30,21)(26,32,31,27)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,7)(6,8)(9,12)(15,16)(18,25)(19,28)(20,30)(27,32)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,15)(12,14)(13,16)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,3)(2,8)(4,13)(5,11)(6,7)(9,15)(10,14)(12,16)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,26,24,31)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,22,23,24)(18,28,25,19)(20,27,30,32)(21,31,29,26)(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=gi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,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, 10050837);