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 | |
4233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,14,16)(4,9,12,13)(5,15,11,6)(17,18)(19,23)(20,24)(21,27)(22,25)(26,28)(29,32)(30,31)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,47,48);(1,3)(2,8)(4,5)(6,13)(7,14)(9,15)(10,16)(11,12)(17,19,22,27)(18,23,25,21)(20,26,29,31)(24,28,32,30)(33,35)(34,39)(36,47)(37,44)(38,45)(40,48)(41,46)(42,43);(1,4)(2,9)(3,11)(5,14)(6,8)(7,12)(10,13)(15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)(33,36,38,44)(34,40,42,46)(35,37,45,47)(39,41,43,48);(1,4)(2,9)(3,5)(6,16)(7,12)(8,15)(10,13)(11,14)(17,20)(18,24)(19,26)(21,30)(22,29)(23,28)(25,32)(27,31)(33,35)(34,39)(36,37)(38,45)(40,41)(42,43)(44,47)(46,48);(1,5)(2,6)(3,12)(4,14)(7,11)(8,9)(10,15)(13,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)(33,37)(34,41)(35,44)(36,45)(38,47)(39,46)(40,43)(42,48);(1,6)(2,5)(3,13)(4,8)(7,15)(9,14)(10,11)(12,16)(17,21)(18,19)(20,28)(22,23)(24,31)(25,27)(26,32)(29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319240);