Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
3277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,8)(13,14)(15,20)(16,24)(17,27)(18,28)(19,21)(22,23)(25,26);(2,3)(10,12)(13,15,19,16)(14,20,21,24)(17,25,28,22)(18,23,27,26);(9,10)(11,12)(13,16,19,15)(14,20,21,24)(17,25,28,22)(18,26,27,23);(2,3)(5,6,7,8)(13,17)(14,18)(15,22)(16,25)(19,28)(20,26)(21,27)(23,24);(1,3,4,2)(13,18)(14,17)(15,23)(16,26)(19,27)(20,25)(21,28)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2828542);