Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,10,12)(7,9,8,11)(13,14)(15,16)(17,20)(18,19)(23,25)(27,28);(5,7,10,8)(6,11,12,9)(13,15,17,19)(14,18,20,16)(21,22,26,24)(23,27,25,28);(1,2,4,3)(13,16)(14,19)(15,20)(17,18)(21,23,26,25)(22,27,24,28);(1,2,4,3)(5,8,10,7)(6,9,12,11)(13,16)(14,15)(17,18)(19,20)(21,24,26,22)(23,28,25,27);(1,3,4,2)(5,9)(6,8)(7,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,25,26,23)(22,28,24,27)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=gh,d^2=fg,e^2=gh,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, 1353641);