Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3488 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,21)(19,22)(23,24);(2,5)(6,8)(10,14)(15,17)(16,21)(22,23);(1,3,4,7)(2,6,5,8)(10,15)(13,20)(14,17)(16,22)(19,24)(21,23);(9,11)(10,16)(12,18)(13,19)(14,21)(15,22)(17,23)(20,24);(11,18)(16,21)(19,24)(22,23)Polycyclic presentation:
2^8|c^2=fh,g^2=h,b^a=bf,c^a=cg,g^a=gh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #60525⋊C3 |
| maximal subgroup of | #249912, #249916, #249920, … |
| maximal normal subgroup of | #249912, #249916, #249920, … |
| maximal quotient of | #6370911, #6371345, #6371390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53513);