Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #52249 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,11,14,8)(5,10,6,9)(12,15,13,16)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3,7,14)(2,8,4,11)(5,12,6,13)(9,15,10,16)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,7,2)(3,11,14,8)(5,9,6,10)(12,15,13,16)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,32)(26,28)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(1,5,7,6)(2,9,4,10)(3,12,14,13)(8,15,11,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(35,42)(36,45)(38,47)(39,43);(1,5,7,6)(2,10,4,9)(3,13,14,12)(8,15,11,16)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,9,4,10)(3,13,14,12)(8,15,11,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012759);