Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,17)(11,18,16,13)(12,20,22,15)(19,23,21,24);(9,11,13,10,14,16,18,17)(12,21,23,20,22,19,24,15);(1,2,3,4)(9,12,14,22)(10,15,17,20)(11,19,16,21)(13,23,18,24);(6,8)(12,22)(15,20)(19,21)(23,24)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6473311, #6473313, #6473314, … |
| maximal normal subgroup of | #6473311, #6473313, #6473314, … |
| maximal quotient of | #351820, #351821, #351824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26176);