Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(1,2)(3,4)(5,6)(7,8);(2,4)(5,7,8,6)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^8|c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #359149, #359150, #359151, … |
| maximal normal subgroup of | #359149, #359150, #359151, … |
| maximal quotient of | #238661, #238665, #238668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16467);