Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(5,7,8,6)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,2,3,4);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #109831, #109832, #109977, … |
| maximal normal subgroup of | #109831, #109832, #109977, … |
| maximal quotient of | #203615, #203625, #203635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14157);