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 | |
4259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,12,16)(11,15,14,13)(17,18,22,24)(19,27,28,23)(20,29,30,25)(21,26,31,32)(33,34,37,41)(35,39,43,38)(36,40,45,44)(42,47,48,46);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(17,19,22,28)(18,23,24,27)(20,21,30,31)(25,26,29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(3,5)(4,12)(6,8)(9,16)(18,24)(20,30)(21,31)(23,27)(33,34,37,41)(35,38,43,39)(36,44,45,40)(42,47,48,46);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20,22,30)(18,25,24,29)(19,21,28,31)(23,26,27,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,21)(18,26)(19,20)(22,31)(23,25)(24,32)(27,29)(28,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,30)(31,32)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=i,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, 8914561);