Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 403 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:
(5,6,8,10)(7,11,12,9)(13,14,18,20)(15,23,16,26)(17,28,24,22)(19,27,21,25);(1,2)(3,4)(13,15,17,25,18,16,24,27)(14,19,22,23,20,21,28,26);(2,4)(5,7,8,12)(6,9,10,11)(14,20)(19,21)(22,28)(23,26);(13,16,17,27,18,15,24,25)(14,21,22,26,20,19,28,23)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=fh,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #503656, #503692, #504125, … |
| maximal normal subgroup of | #503656, #503692, #504125, … |
| maximal quotient of | #305771, #305779, #305787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19050);