Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2617 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(10,12)(11,15)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31);(6,8)(11,15)(14,16)(17,19,22,20)(18,23,26,24)(21,28,30,27)(25,32,29,31);(5,6)(7,8)(9,10,13,12)(11,14,15,16)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31);(1,2)(3,4)(9,11)(10,14)(12,16)(13,15)(17,18)(19,24)(20,23)(21,29)(22,26)(25,30)(27,31)(28,32);(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2976876);