Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,8)(13,14,18,21)(15,17,23,19)(16,24,27,20)(22,26,25,28);(2,3)(10,12)(13,15,18,23)(14,19,21,17)(16,25,27,22)(20,28,24,26);(1,2,4,3)(9,10,11,12)(13,14)(15,19)(16,20)(17,23)(18,21)(22,26)(24,27)(25,28);(5,6)(7,8)(13,16,18,27)(14,20,21,24)(15,22,23,25)(17,28,19,26);(1,3,4,2)(13,17,18,19)(14,15,21,23)(16,26,27,28)(20,25,24,22)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3944321);