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 | |
4217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,4,11,8)(6,13,14,9)(10,15,16,12)(18,23)(19,26)(25,31)(27,29)(33,34)(35,41)(36,40)(37,38)(39,42)(43,48)(44,47)(45,46);(1,3,7,11)(2,8,5,4)(6,10,14,16)(9,15,13,12)(17,18,22,23)(19,24,26,21)(20,25,28,31)(27,32,29,30)(35,42)(39,41)(43,44)(47,48);(2,5)(4,8)(6,14)(10,16)(19,26)(20,28)(21,24)(25,31)(33,35,38,42)(34,39,37,41)(36,44,46,43)(40,48,45,47);(1,4)(2,3)(5,11)(6,12)(7,8)(9,10)(13,16)(14,15)(17,19)(18,24)(20,29)(21,23)(22,26)(25,30)(27,28)(31,32)(33,36,38,46)(34,40,37,45)(35,43,42,44)(39,47,41,48);(1,5,7,2)(3,8,11,4)(6,13,14,9)(10,15,16,12)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,35)(34,41)(36,44)(37,39)(38,42)(40,47)(43,46)(45,48);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,21,22,24)(18,19,23,26)(20,30,28,32)(25,27,31,29)(33,37,38,34)(35,39,42,41)(36,45,46,40)(43,47,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=i,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900306);