Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 262144 |
| 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 | C2×#525 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24);(5,7,8,6)(10,15)(12,16)(18,21)(20,23);(1,2)(3,4)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #105900, #129329, #129347, … |
| maximal normal subgroup of | #105900, #129329, #129347, … |
| maximal quotient of | #513548, #513570, #513574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27189);