Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
747 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)(7,8)(9,10)(11,18)(12,17)(13,15)(14,21)(16,19)(20,24)(22,23);(1,2,3,4)(10,15)(16,23)(17,21)(18,24);(9,11)(10,16)(12,19)(13,20)(14,22)(15,23)(17,24)(18,21);(6,8)(9,12,13,14)(10,17,15,21)(11,19,20,22)(16,24,23,18)Polycyclic presentation:
2^8|b^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6470257, #6470258, #6470267, … |
| maximal normal subgroup of | #6470257, #6470258, #6470267, … |
| maximal quotient of | #347520, #347523, #347538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25898);