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 | |
4197 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)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,34,38,39)(35,37,41,46)(36,40,45,47)(42,44,43,48);(2,6)(8,9)(10,13)(15,16)(21,29)(24,31)(27,28)(30,32)(34,39)(37,46)(40,47)(44,48);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,35,38,41)(34,37,39,46)(36,43,45,42)(40,48,47,44);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36,38,45)(34,40,39,47)(35,42,41,43)(37,44,46,48);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,43)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8809703);