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 | #53533 | C2 |
| 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,14)(15,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,4,6,10)(2,8,9,5)(3,12,13,16)(7,15,11,14)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(2,9)(4,10)(7,11)(12,16)(18,24)(19,27)(20,29)(30,31)(36,44)(39,48)(41,43)(45,47);(3,13)(7,11)(12,16)(14,15)(18,24)(19,27)(23,28)(26,32)(35,42)(38,46)(41,43)(45,47);(1,5,6,8)(2,10,9,4)(3,14,13,15)(7,16,11,12)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,f^2=gh,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, 8009168);