Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34925 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,11)(10,12)(13,14,18,21)(15,22,24,19)(16,25,17,20)(23,27,26,28);(2,3)(13,15,18,24)(14,19,21,22)(16,23,17,26)(20,27,25,28);(1,2,4,3)(6,9)(7,12)(10,11);(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,22,28);(1,3,4,2)(13,17,18,16)(14,20,21,25)(15,23,24,26)(19,28,22,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719003);