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 |
| prime radical | #49245 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,14,16)(10,12,13,15)(17,18,23,24)(19,21,26,31)(20,25,29,32)(22,30,28,27)(33,34,38,39)(35,43,44,40)(36,41,45,48)(37,46,47,42);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,19)(18,21)(20,28)(22,29)(23,26)(24,31)(25,27)(30,32)(34,39)(36,45)(40,43)(42,46);(18,24)(19,26)(20,29)(27,30)(33,35,38,44)(34,40,39,43)(36,42,45,46)(37,41,47,48);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,16,13)(17,20,23,29)(18,25,24,32)(19,28,26,22)(21,27,31,30)(33,35)(34,40)(36,46)(37,41)(38,44)(39,43)(42,45)(47,48);(1,3)(2,4)(5,13)(6,11)(7,9)(8,15)(10,14)(12,16)(17,21)(18,26)(19,24)(20,30)(22,32)(23,31)(25,28)(27,29)(33,36)(34,41)(35,42)(37,43)(38,45)(39,48)(40,47)(44,46);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22,23,28)(18,27,24,30)(19,29,26,20)(21,25,31,32)(33,37,38,47)(34,42,39,46)(35,41,44,48)(36,40,45,43)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=gh,f^2=h,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, 10015856);