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 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18)(19,21)(20,25)(22,30)(23,24)(26,31)(27,32)(28,29)(37,45)(40,46)(42,47)(43,48);(2,6)(7,9)(8,12)(14,15)(17,19)(18,21)(20,23)(22,27)(24,25)(26,29)(28,31)(30,32)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,46)(47,48);(2,6)(7,9)(8,12)(14,15)(18,24)(19,20)(26,30)(27,28)(34,39)(35,41)(40,46)(42,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29)(33,35,38,41)(34,36,39,44)(37,42,45,47)(40,43,46,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,25)(18,20,24,19)(22,29,31,32)(26,28,30,27)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066730);