Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1119 conjugacy classes of subgroups; only the 719 normal subgroups are shown.
This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 719 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 — 5 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(14,18)(16,24)(19,21)(22,28);(6,9)(10,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(5,6,8,9)(7,10,11,12);(5,7)(6,10)(8,11)(9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107223, #107302, #107340, … |
| maximal normal subgroup of | #107223, #107302, #107340, … |
| maximal quotient of | #513784, #513803, #513831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27353);