Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,12)(15,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,34)(35,37)(36,43)(38,40)(39,41)(42,45)(44,47)(46,48);(2,5)(6,8)(10,14)(15,16)(18,19)(20,27)(24,26)(28,32)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5238444);