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 | |
4169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,6)(5,13,15,10)(11,14,12,16)(20,28)(24,31)(27,32)(29,30)(34,37)(36,40)(41,45)(44,46);(2,8)(3,4)(10,13)(11,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,35)(37,43)(38,46)(39,40)(41,48)(42,45)(44,47);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,34)(35,36,43,40)(38,45,47,41)(42,44,48,46);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,46)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8366299);