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×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3781 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(3,9)(8,12)(10,15)(17,18)(19,25)(20,22)(21,28)(23,26)(24,29)(27,30)(31,32)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,2,6,5)(3,7,9,11)(4,12,13,8)(10,16,15,14)(18,20)(19,26)(24,28)(27,32)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(3,9)(7,11)(10,15)(14,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,4,6,13)(2,8,5,12)(3,10,9,15)(7,14,11,16)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46);(1,5,6,2)(3,11,9,7)(4,12,13,8)(10,16,15,14)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022604);