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 | |
4126 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)(3,9)(4,7)(5,13)(6,8)(10,14)(11,15)(12,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34,38,37)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(35,42)(39,41)(43,48)(46,47);(2,8)(7,9)(10,13)(15,16)(22,30)(25,28)(27,29)(31,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,36,38,45)(34,40,37,44)(35,43,42,48)(39,46,41,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,37,38,34)(35,39,42,41)(36,44,45,40)(43,46,48,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=g,e^2=gh,f^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, 8087559);