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) = C27 |
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 | |
2221 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)(5,6,8,9)(7,11,12,10)(13,14,18,20)(15,16,17,19)(21,22)(23,26)(24,28)(25,27);(6,9)(7,12)(13,15)(14,19)(16,20)(17,18)(21,23,25,28)(22,26,27,24);(6,9)(10,11)(13,16,18,19)(14,17,20,15)(21,24)(22,23)(25,26)(27,28);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,17,18,15)(14,16,20,19);(5,6)(7,11)(8,9)(10,12)(13,16)(14,17)(15,20)(18,19)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=h,d^2=fhi,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, 4009165);