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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,12)(6,9)(7,8)(10,11)(13,14)(15,16)(18,23)(19,25)(24,30)(26,31)(33,34,38,41)(35,39,44,47)(36,37,40,45)(42,43,46,48);(1,3,7,5)(2,4,8,12)(6,10,15,13)(9,11,16,14)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(2,8)(3,5)(9,16)(10,13)(20,27)(24,30)(26,31)(28,32)(33,35,38,44)(34,39,41,47)(36,42,40,46)(37,43,45,48);(1,4)(2,3)(5,8)(6,14)(7,12)(9,13)(10,16)(11,15)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36)(34,37)(35,42)(38,40)(39,43)(41,45)(44,46)(47,48);(1,5,7,3)(2,4,8,12)(6,13,15,10)(9,11,16,14)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,6)(2,9)(3,10)(4,11)(5,13)(7,15)(8,16)(12,14)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37)(34,40)(35,43)(36,41)(38,45)(39,46)(42,47)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8436151);