Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(10,13)(11,15)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,29)(30,32);(2,4)(6,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,21)(20,22)(23,24)(26,29)(27,32);(3,7)(6,8)(10,13)(14,16)(18,23)(19,20)(25,26,28,29)(27,30,32,31);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,24,23,21)(25,26)(27,30)(28,29)(31,32);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21,22,24)(18,19,23,20)(25,27,28,32)(26,30,29,31)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4428149);