Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14,18,16)(15,17,20,19)(21,22,26,25)(23,28,24,27);(6,7)(10,11)(13,15)(14,19)(16,17)(18,20)(22,25)(23,24);(8,12)(10,11)(15,20)(17,19)(21,23,26,24)(22,27,25,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16,18,14)(15,19,20,17)(21,24,26,23)(22,28,25,27);(5,8,9,12)(6,10,7,11)(13,17,18,19)(14,15,16,20)(21,25,26,22)(23,27,24,28)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4029913);