Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2659 conjugacy classes of subgroups; only the 999 normal subgroups are shown.
This group has 2659 conjugacy classes of subgroups — too many to draw. Only its 999 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(10,11)(13,21)(15,17)(19,24);(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22);(1,2)(3,4)(5,6)(7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| Fitting subgroup of | #34174⋊C3 |
| maximal subgroup of | #105840, #106773, #106991, … |
| maximal normal subgroup of | #105840, #106773, #106991, … |
| maximal quotient of | #513548, #513678, #513717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27162);