Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 196608 |
| 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×#548 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(9,10,15,13)(11,18,12,16)(14,21,22,17)(19,23,20,24);(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24);(5,6,7,8)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24);(9,13,15,10)(11,16,12,18)(14,21,22,17)(19,23,20,24);(1,3,4,2)(9,14,15,22)(10,17,13,21)(11,19,12,20)(16,23,18,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| Fitting subgroup of | #34312⋊C3 |
| maximal subgroup of | #6383826, #6383830, #6383833, … |
| maximal normal subgroup of | #6383826, #6383830, #6383833, … |
| maximal quotient of | #513630, #513658, #513695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27300);