Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| 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 | C2×#547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,5)(3,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23);(11,19)(14,17)(18,24)(21,23);(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^8|a^2=h,c^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,g^c=gh61 conjugacy classes — view the full character table.
| maximal subgroup of | #245532, #245535, #245556, … |
| maximal normal subgroup of | #245532, #245535, #245556, … |
| maximal quotient of | #6386134, #6387793, #6388179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54683);