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 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,13,8)(4,14,11,6)(5,10,15,16)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(33,34,39,42)(35,40,38,48)(36,41,46,37)(43,44,45,47);(1,3)(2,8)(4,5)(6,10)(7,13)(9,12)(11,15)(14,16)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(2,9)(3,13)(4,11)(10,16)(18,24)(19,27)(20,29)(26,30)(33,36,39,46)(34,37,42,41)(35,43,38,45)(40,47,48,44);(1,4,7,11)(2,6,9,14)(3,5,13,15)(8,10,12,16)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,5)(2,10)(3,11)(4,13)(6,12)(7,15)(8,14)(9,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,6)(2,11)(3,10)(4,9)(5,12)(7,14)(8,15)(13,16)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=hi,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, 10102936);