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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,14)(4,9,13,16)(5,6,10,15)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,38,42)(35,43,44,39)(36,40,45,48)(37,41,46,47);(1,3,7,12)(2,8,11,14)(4,6,13,15)(5,9,10,16)(18,19)(23,25)(24,26)(30,31)(35,44)(36,45)(39,43)(40,48);(3,12)(4,13)(8,14)(9,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,38,44)(34,39,42,43)(36,46,45,37)(40,41,48,47);(1,4)(2,9)(3,6)(5,14)(7,13)(8,10)(11,16)(12,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,45)(34,40,42,48)(35,37,44,46)(39,47,43,41);(1,5)(2,6)(3,9)(4,14)(7,10)(8,13)(11,15)(12,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,34)(35,39)(36,40)(37,47)(38,42)(41,46)(43,44)(45,48);(1,6)(2,10)(3,13)(4,12)(5,11)(7,15)(8,16)(9,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,41)(35,36)(38,46)(39,48)(40,43)(42,47)(44,45)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619572);