Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C23×C8 | C22 |
| socle | C23 | C22⋊C8 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 44 points:
(5,6)(7,11,12,9)(8,10)(13,14,17,22,18,23,32,40,19,24,33,41,16,21,30,38)(15,25,27,42,26,36,43,39,28,37,31,44,29,20,34,35);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,29)(17,31)(18,26)(19,28)(21,37)(22,39)(23,25)(24,36)(27,33)(30,43)(32,34)(35,41)(38,42)(40,44)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,g^2=h,b^a=bc,c^a=cf,c^b=ch,d^b=dfh76 conjugacy classes — view the full character table.
| maximal subgroup of | #93903, #93906, #93952, … |
| maximal normal subgroup of | #93903, #93906, #93952, … |
| maximal quotient of | #92987, #92988, #92989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 174);