Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(10,13)(14,16)(17,18,21,19)(20,22,24,23);(2,4)(5,6,7,8)(10,13)(11,15)(18,19)(20,24);(1,2,3,4)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,22,24,23);(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,d^2=ef,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de52 conjugacy classes — view the full character table.
| Fitting subgroup of | #14678⋊C3 |
| maximal subgroup of | #192397, #192398, #192413, … |
| maximal normal subgroup of | #192397, #192398, #192413, … |
| maximal quotient of | #154058, #154073, #155675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7666);