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) = C25 |
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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,20)(12,21)(13,17)(14,18)(15,19)(16,23)(22,24);(5,6)(7,8)(9,11,13,21,15,16,14,22)(10,12,18,20,19,24,17,23);(2,4)(6,8)(9,12)(10,16)(11,19)(13,23)(14,20)(15,24)(17,22)(18,21);(9,13,15,14)(10,17,19,18)(11,21,16,22)(12,23,24,20)Polycyclic presentation:
2^8|a^2=ef,b^2=e,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^c=eh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #359414, #359415, #359416, … |
| maximal normal subgroup of | #359414, #359415, #359416, … |
| maximal quotient of | #242397, #242402, #242407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16597);