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 | #58169 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4231 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)(3,11)(4,5)(6,9)(7,10)(8,15)(12,14)(13,16)(17,18)(19,25)(20,22)(21,28)(23,26)(24,30)(27,31)(29,32)(33,34)(35,38)(36,40)(37,39)(41,46)(42,45)(43,48)(44,47);(1,3,7,15)(2,8,10,11)(4,12,9,16)(5,13,6,14)(17,19)(18,23)(20,25)(21,29)(22,26)(24,32)(27,30)(28,31)(33,35,37,41)(34,38,39,46)(36,43,44,42)(40,48,47,45);(1,4,7,9)(2,5,10,6)(3,12,15,16)(8,13,11,14)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31)(34,39)(35,41)(36,44)(45,48);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,16,11,12)(18,20)(19,26)(24,28)(27,29)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,5,7,6)(2,4,10,9)(3,14,15,13)(8,16,11,12)(17,21,22,30)(18,24,20,28)(19,27,26,29)(23,31,25,32)(34,39)(35,41)(40,47)(42,43);(1,6,7,5)(2,9,10,4)(3,14,15,13)(8,16,11,12)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gh,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, 8012822);