Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3094 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)(13,14,18,22)(15,16,20,19)(17,25,27,21)(23,26,28,24);(10,12)(13,15,18,20)(14,19,22,16)(17,23,27,28)(21,26,25,24);(1,2,4,3)(9,10,11,12)(13,15)(14,19)(16,22)(17,23)(18,20)(21,26)(24,25)(27,28);(5,6)(7,8)(13,16,18,19)(14,20,22,15)(17,26,27,24)(21,23,25,28);(1,3,4,2)(13,17,18,27)(14,21,22,25)(15,23,20,28)(16,24,19,26)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1301288);