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 | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1473 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)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22);(5,6,8,7)(10,12)(15,17)(16,20)(22,23);(1,2)(3,4)(5,7,8,6)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=h,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 | #96970, #129349, #146672, … |
| maximal normal subgroup of | #96970, #129349, #146672, … |
| maximal quotient of | #513570, #513574, #513605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27254);