Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ 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:
(2,4)(5,6,10,12)(7,9,8,11)(14,16)(15,19);(5,7,10,8)(6,11,12,9)(21,22,24,26)(23,27,28,25);(5,7,10,8)(6,9,12,11)(13,14,17,16)(15,18,19,20)(21,23,24,28)(22,25,26,27);(1,2,3,4)(5,8,10,7)(6,9,12,11)(13,15)(14,18)(16,20)(17,19);(5,9)(6,8)(7,12)(10,11)(13,16,17,14)(15,20,19,18)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1402064);