Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C2×#392 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
895 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 895 conjugacy classes of subgroups — too many to draw. Only its 151 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,19,20,16)(21,22)(23,25)(24,27)(26,28);(7,11)(9,12)(14,18)(15,20)(21,23,24,28)(22,25,27,26);(5,7)(6,9)(8,11)(10,12)(13,15,17,20)(14,16,18,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^8|a^2=efg,b^2=ef,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| Fitting subgroup of | #17149⋊C3 |
| maximal subgroup of | C2×#17149, #844933, #845009, … |
| maximal normal subgroup of | C2×#17149, #844933, #845009, … |
| maximal quotient of | #156474, #156501, #156502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10137);