Finite Group DatabasebetaFinite Group Databasebeta| order | 608 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C19 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 152 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,26)(23,24)(25,27);(21,24)(25,26);(20,21)(22,25)(23,24)(26,27);(20,22)(21,25)(23,27)(24,26)323 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(608, 192);