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 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,10,4)(5,7,13,14)(11,15,16,12)(17,18)(19,26)(20,28)(21,25)(22,23)(24,29)(27,31)(30,32)(33,34,38,43)(35,41,44,39)(36,40,42,37)(45,46,47,48);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,23,25)(18,21,22,26)(20,27,29,32)(24,31,28,30)(35,44)(39,41)(45,47)(46,48);(3,10)(4,9)(5,13)(7,14)(18,22)(20,29)(21,26)(27,32)(33,35,38,44)(34,39,43,41)(36,47,42,45)(37,48,40,46);(1,3,6,10)(2,4,8,9)(5,11,13,16)(7,12,14,15)(17,20,23,29)(18,24,22,28)(19,27,25,32)(21,31,26,30)(33,36,38,42)(34,40,43,37)(35,45,44,47)(39,48,41,46);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,23,26)(18,25,22,19)(20,30,29,31)(24,27,28,32)(33,35)(34,41)(36,47)(37,46)(38,44)(39,43)(40,48)(42,45);(1,5)(2,7)(3,11)(4,12)(6,13)(8,14)(9,15)(10,16)(17,22,23,18)(19,26,25,21)(20,28,29,24)(27,30,32,31)(33,37,38,40)(34,42,43,36)(35,46,44,48)(39,47,41,45)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=i,d^2=gi,e^2=hi,f^2=h,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, 8924149);