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 | |
4305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(18,20)(19,25)(24,28)(26,31)(33,34,37,39)(35,41,43,38)(36,44,46,40)(42,47,45,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,35,37,43)(34,38,39,41)(36,45,46,42)(40,48,44,47);(2,6)(7,10)(8,12)(14,16)(19,25)(23,27)(26,31)(30,32)(34,39)(38,41)(40,44)(47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32)(33,34)(35,38)(36,44)(37,39)(40,46)(41,43)(42,48)(45,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,36,37,46)(34,40,39,44)(35,42,43,45)(38,47,41,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,47)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731951);