Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C32 · Schur multiplier (H2) = C214×C3 |
| direct product | C32×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C26 |
| commutator, Frattini, cosocle | C2 | C26×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
17388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2,3)(7,8)(9,14)(10,15)(11,23)(12,13)(16,24)(17,27)(18,32)(19,25)(20,29)(21,35)(22,26)(28,33)(30,36)(31,38)(34,37);(4,5,6)(8,13)(14,25)(15,26)(16,23)(27,29)(28,32)(30,35)(37,38);(7,9,12,19)(8,14,13,25)(10,20,22,17)(11,18,24,33)(15,29,26,27)(16,28,23,32)(21,34,36,31)(30,38,35,37);(7,10,12,22)(8,15,13,26)(9,17,19,20)(11,21,24,36)(14,27,25,29)(16,30,23,35)(18,31,33,34)(28,37,32,38);(7,8)(9,14)(10,15)(11,16)(12,13)(17,27)(18,28)(19,25)(20,29)(21,30)(22,26)(23,24)(31,37)(32,33)(34,38)(35,36);(7,11)(8,16)(9,18)(10,21)(12,24)(13,23)(14,28)(15,30)(17,31)(19,33)(20,34)(22,36)(25,32)(26,35)(27,37)(29,38)585 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4655);