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 | |
4203 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18,23,25)(19,26,20,24)(21,31,32,22)(27,28,29,30)(33,34)(35,39)(36,41)(37,45)(38,40)(42,46)(43,44)(47,48);(2,7)(3,10)(9,14)(11,16)(17,19,23,20)(18,24,25,26)(21,27,32,29)(22,28,31,30)(33,35,38,41)(34,39,40,36)(37,43,46,48)(42,47,45,44);(5,13)(9,14)(11,16)(12,15)(18,25)(22,31)(24,26)(28,30)(34,40)(35,41)(42,45)(43,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,23,19)(18,26,25,24)(21,29,32,27)(22,30,31,28)(33,36,38,39)(34,41,40,35)(37,44,46,47)(42,48,45,43);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,37,38,46)(34,42,40,45)(35,43,41,48)(36,44,39,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8086850);