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 | |
4230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,13,16)(4,10,14,15)(5,11,12,6)(17,18,22,26)(19,21,25,23)(20,24,30,32)(27,29,28,31)(33,34)(35,42)(36,37)(38,39)(40,45)(41,48)(43,44)(46,47);(1,3,7,13)(2,8,9,16)(4,12,14,5)(6,15,11,10)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(34,39)(36,41)(40,42)(43,46);(2,9)(3,13)(4,14)(6,11)(19,25)(21,23)(27,28)(29,31)(33,35,38,45)(34,40,39,42)(36,43,41,46)(37,47,48,44);(1,4)(2,10)(3,12)(5,13)(6,8)(7,14)(9,15)(11,16)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,29,32,31)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5,7,12)(2,11,9,6)(3,14,13,4)(8,15,16,10)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,29,23,31)(33,36,38,41)(34,37,39,48)(35,43,45,46)(40,47,42,44);(1,6,7,11)(2,12,9,5)(3,15,13,10)(4,8,14,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,28,32,27)(33,37)(34,41)(35,44)(36,39)(38,48)(40,43)(42,46)(45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831869);