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 | |
4048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,23,24)(19,21,20,25)(22,30,31,26)(27,32,28,29)(33,34,38,42)(35,41,37,39)(36,40,45,48)(43,47,44,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,21)(20,23)(22,27)(24,25)(26,29)(28,31)(30,32)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(2,6)(7,9)(8,13)(14,15)(18,24)(19,20)(26,30)(27,28)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,25)(18,19,24,20)(22,29,31,32)(26,27,30,28)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,31)(18,26,24,30)(19,27,20,28)(21,29,25,32)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 8019563);