Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
387 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,21,18,22,16,17)(15,23,24,25,26,20,27,28);(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,20,22,23)(16,27,19,24)(17,25,21,28);(1,2,3,4)(13,16,18,19)(14,21,22,17)(15,24,26,27)(20,25,23,28);(5,8,9,7)(6,10,11,12)(13,17,16,22,18,21,19,14)(15,25,27,23,26,28,24,20)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=eg,g^2=h,b^a=be,c^a=cf,c^b=ch,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #122433, #124817, #128997, … |
| maximal normal subgroup of | #122433, #124817, #128997, … |
| maximal quotient of | #213462, #213463, #213464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14706);