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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,10,13,7)(4,12,14,9)(5,15,16,11)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(33,34)(35,37)(36,44)(38,41)(39,43)(40,45)(42,46)(47,48);(1,3,6,13)(2,7,8,10)(4,11,14,15)(5,12,16,9)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(2,8)(4,14)(7,10)(11,15)(18,24)(19,27)(20,29)(26,30)(36,45)(40,44)(42,48)(46,47);(1,4,6,14)(2,9,8,12)(3,11,13,15)(5,10,16,7)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32)(33,36,38,45)(34,40,41,44)(35,42,43,48)(37,47,39,46);(1,3)(2,10)(4,15)(5,9)(6,13)(7,8)(11,14)(12,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25)(33,37,38,39)(34,35,41,43)(36,46,45,47)(40,48,44,42);(1,5)(2,11)(3,12)(4,10)(6,16)(7,14)(8,15)(9,13)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)(33,35)(34,37)(36,42)(38,43)(39,41)(40,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9034759);