Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1847 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)(6,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(1,2,3,4)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(5,6,7,8)(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)Polycyclic presentation:
2^8|b^2=f,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| Fitting subgroup of | #34641⋊C3 |
| maximal subgroup of | #111253, #111271, #112625, … |
| maximal normal subgroup of | #111253, #111271, #112625, … |
| maximal quotient of | #515054, #515154, C4×#2656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27629);