Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,18)(15,20);(1,2,4,3)(5,6,9,8)(7,11,12,10)(13,14,17,18)(15,19,20,16)(21,22,24,26)(23,27,28,25);(1,3,4,2)(6,8)(10,11)(14,18)(16,19)(21,23,24,28)(22,25,26,27);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,18)(17,20);(5,8,9,6)(7,11,12,10)(13,16,17,19)(14,15,18,20)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 484421);