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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(14,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(10,13)(14,16)(18,19)(20,27)(24,26)(28,32)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,34)(35,36)(37,40)(38,41)(39,44)(42,43)(45,47)(46,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=h,d^2=f,e^2=hi,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, 5270468);