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×#2676 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3339 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)(13,14,17,21)(15,19,23,18)(16,20,26,25)(22,28,24,27);(2,3)(10,12)(13,15,17,23)(14,18,21,19)(16,22,26,24)(20,27,25,28);(1,2,4,3)(9,10)(11,12)(13,15)(14,19)(16,24)(17,23)(18,21)(20,27)(22,26)(25,28);(5,6)(7,8)(13,14)(15,19)(16,25)(17,21)(18,23)(20,26)(22,27)(24,28);(1,3,4,2)(13,16)(14,20)(15,22)(17,26)(18,27)(19,28)(21,25)(23,24)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=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, 3941424);