Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1030 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)(10,15)(11,19)(12,14)(16,23)(20,21)(22,24);(2,5)(6,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23);(9,11,13,20)(10,16,15,22)(12,19,14,21)(17,24,18,23);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,17,15,18)(11,19,20,21)(16,24,22,23);(10,15)(16,22)(17,18)(23,24)Polycyclic presentation:
2^8|c^2=h,d^2=f,g^2=h,b^a=bf,c^a=cg,d^a=dh,g^a=gh,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6439758, #6439760, #6439763, … |
| maximal normal subgroup of | #6439758, #6439760, #6439763, … |
| maximal quotient of | #6370989, #6371799, #6371801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53564);