Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
651 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 651 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(7,11)(10,12)(13,14)(15,17)(16,19)(18,20)(21,22,24,26)(23,27,28,25);(6,9)(10,12)(14,15)(18,19)(21,22,24,26)(23,25,28,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,22)(23,25)(24,26)(27,28);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^8|a^2=ef,b^2=ef,c^2=e,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| Fitting subgroup of | #15171⋊C3 |
| maximal subgroup of | C2×#15171, #608430, #608438, … |
| maximal normal subgroup of | C2×#15171, #608430, #608438, … |
| maximal quotient of | #154182, #154409, #154410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8159);