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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,11,13,8)(5,9,6,10)(12,16,14,15)(17,18,22,23)(19,27,29,24)(20,26,21,25)(28,31,30,32)(33,34)(35,38)(36,40)(37,39)(41,46)(42,45)(43,48)(44,47);(1,3,7,13)(2,8,4,11)(5,14,6,12)(9,16,10,15)(18,23)(20,21)(24,27)(28,30)(33,35,37,41)(34,38,39,46)(36,43,44,42)(40,48,47,45);(1,4,7,2)(3,11,13,8)(5,10,6,9)(12,16,14,15)(17,19,22,29)(18,24,23,27)(20,30,21,28)(25,32,26,31)(34,39)(35,41)(36,44)(45,48);(2,4)(8,11)(9,10)(15,16)(17,20,22,21)(18,25,23,26)(19,28,29,30)(24,31,27,32)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,5,7,6)(2,9,4,10)(3,12,13,14)(8,15,11,16)(17,21,22,20)(18,26,23,25)(19,28,29,30)(24,31,27,32)(34,39)(35,41)(40,47)(42,43);(1,6,7,5)(2,10,4,9)(3,12,13,14)(8,15,11,16)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=h,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, 8012819);