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 | |
4219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,6)(5,13,15,10)(11,14,12,16)(17,18,22,25)(19,24,26,23)(20,29,30,21)(27,28,32,31)(33,34)(35,44)(36,46)(37,39)(38,48)(40,43)(41,47)(42,45);(2,8)(3,4)(10,13)(11,12)(19,26)(23,24)(27,32)(28,31)(33,35)(34,36)(37,48)(38,41)(39,45)(40,46)(42,47)(43,44);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,25)(29,31)(30,32)(33,36)(34,35)(37,45)(38,47)(39,48)(40,44)(41,42)(43,46);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,19)(18,24)(20,27)(21,31)(22,26)(23,25)(28,29)(30,32)(33,37,40,47)(34,41,43,39)(35,42,46,48)(36,45,44,38);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,20)(18,21)(19,27)(22,30)(23,28)(24,31)(25,29)(26,32)(33,38,40,45)(34,42,43,48)(35,41,46,39)(36,47,44,37);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,21)(18,20)(19,28)(22,29)(23,27)(24,32)(25,30)(26,31)(33,39,40,41)(34,37,43,47)(35,45,46,38)(36,48,44,42)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332561);