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 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(18,23)(20,27)(21,29)(26,31)(33,34)(35,39)(36,45)(37,47)(38,42)(40,46)(41,43)(44,48);(2,6)(7,12)(8,13)(14,16)(18,23)(19,25)(20,27)(28,32)(33,34,38,42)(35,39,44,48)(36,40,46,45)(37,41,43,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35,38,44)(34,39,42,48)(36,43,46,37)(40,47,45,41);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,48)(42,45);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37,38,43)(34,41,42,47)(35,36,44,46)(39,40,48,45)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8016371);