Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(10,14)(12,20)(15,17)(18,24);(5,7,8,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24);(1,2,3,4);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #105769, #105886, #106093, … |
| maximal normal subgroup of | #105769, #105886, #106093, … |
| maximal quotient of | #513552, #513576, #513594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27197);