Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,4)(7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(34,35)(36,43)(37,45)(47,48);(2,6)(7,8)(18,23)(19,25)(24,31)(26,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,48,46,47);(1,2,5,6)(3,7,4,8)(9,11,13,10)(12,15,16,14)(17,19,22,25)(18,20,23,28)(21,26,30,32)(24,27,31,29)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,48,46,47);(1,3,5,4)(2,7,6,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,29)(22,28)(23,25)(24,32)(26,31)(27,30)(33,36)(34,39)(35,41)(37,46)(38,43)(40,48)(42,47)(44,45);(1,4,5,3)(2,8,6,7)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37,38,45)(34,40,35,42)(36,44,43,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=fgi,d^2=fgh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5648256);