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 | |
4219 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(2,7)(3,12)(9,14)(10,16)(19,28)(20,29)(24,30)(25,32)(34,39)(35,42)(41,44)(43,48);(17,19)(18,24)(20,21)(22,25)(23,28)(26,29)(27,30)(31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,15)(13,16)(17,20)(18,25)(19,21)(22,24)(23,29)(26,28)(27,32)(30,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4)(2,9)(3,10)(5,11)(6,13)(7,14)(8,15)(12,16)(17,21)(18,22)(19,20)(23,26)(24,25)(27,31)(28,29)(30,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,6,15)(2,10,7,16)(3,9,12,14)(4,8,13,11)(17,22)(18,26)(19,25)(20,30)(21,27)(23,31)(24,29)(28,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=i,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8598202);