Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 171 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)(7,10)(8,9)(11,12)(13,14,19,22)(15,23,25,24)(16,18,21,17)(20,27,28,26);(6,9)(10,12)(13,15,18,27,19,25,17,26)(14,20,16,24,22,28,21,23);(5,7,8,11)(6,10,9,12)(13,16,18,22,19,21,17,14)(15,24,27,28,25,23,26,20);(1,3,4,2)(15,25)(20,28)(23,24)(26,27)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=eh,d^2=gh,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #219562, #219564, #219572, … |
| maximal normal subgroup of | #219562, #219564, #219572, … |
| maximal quotient of | #212686, #212689, #212697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14633);