Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C225 |
| direct product | C22×#8108057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2794816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 8 generators on 28 points:
(13,14)(15,19)(16,18)(17,20)(21,22)(23,24)(25,27)(26,28);(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,17,20,19)(21,23)(22,24)(25,26)(27,28);(13,14)(15,17)(16,18)(19,20)(21,22)(23,28)(24,26)(25,27);(3,4);(7,12)(9,11)(15,20)(17,19)(23,26)(24,28);(1,2)(3,4);(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,27)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,24)(22,26)(23,27)(25,28)368 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((13,14)(15,19)(16,18)(17,20)(21,22)(23,24)(25,27)(26,28), (5,6,8,10)(7,11,12,9)(13,14,16,18)(15,17,20,19)(21,23)(22,24)(25,26)(27,28), (13,14)(15,17)(16,18)(19,20)(21,22)(23,28)(24,26)(25,27), (3,4), (7,12)(9,11)(15,20)(17,19)(23,26)(24,28), (1,2)(3,4), (5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,27)(26,28), (5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,24)(22,26)(23,27)(25,28));