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 | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3978 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)(3,8,11,15)(4,6,13,16)(5,10,12,14)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29)(33,34,38,39)(35,43,44,40)(36,41,45,48)(37,46,47,42);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(34,39)(36,45)(40,43)(42,46);(2,9)(3,11)(4,13)(10,14)(33,35,38,44)(34,40,39,43)(36,42,45,46)(37,41,47,48);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32)(33,35)(34,40)(36,46)(37,41)(38,44)(39,43)(42,45)(47,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36)(34,41)(35,42)(37,43)(38,45)(39,48)(40,47)(44,46);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,47)(34,42,39,46)(35,41,44,48)(36,40,45,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,e^2=i,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, 10065837);