Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 195 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:
(1,2,4,3)(5,6)(7,12)(8,10)(9,11)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,27,26,23);(5,7,9,8)(6,10,11,12)(14,18)(19,26)(20,25)(21,28);(13,15,17,23,16,22,24,27)(14,19,20,21,18,26,25,28);(1,3,4,2)(5,8,9,7)(6,10,11,12)(15,22)(19,26)(21,28)(23,27)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=f,d^2=eg,f^2=h,b^a=be,c^a=cf,f^a=fh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124641, #124647, #124803, … |
| maximal normal subgroup of | #124641, #124647, #124803, … |
| maximal quotient of | #213369, #213379, #213390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14716);