Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | C4⋊#401 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
553 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 553 conjugacy classes of subgroups — too many to draw. Only its 199 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,8)(10,11)(13,14,18,16)(15,20,17,19)(21,22,25,28)(23,26,27,24);(5,6,9,8)(7,11,12,10)(13,15,18,17)(14,19,16,20)(21,23,25,27)(22,26,28,24);(5,7,9,12)(6,10,8,11)(13,16,18,14)(15,20,17,19)(21,22)(23,26)(24,27)(25,28);(1,2,3,4)(5,8,9,6)(7,11,12,10)(13,17,18,15)(14,20,16,19)(21,24,25,26)(22,27,28,23)Polycyclic presentation:
2^8|a^2=efg,b^2=ef,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| Fitting subgroup of | C4⋊(#401⋊C3) |
| maximal subgroup of | #401973, #401974, #401977, … |
| maximal normal subgroup of | #401973, #401974, #401977, … |
| maximal quotient of | #154173, #154388, #154397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8142);