Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,22)(17,23)(19,21)(20,26)(24,27)(25,28);(6,9)(10,12)(13,15,18,14)(16,24,26,28)(17,23,19,21)(20,27,22,25);(2,4)(7,11)(10,12)(13,16,18,26)(14,20,15,22)(17,27,19,25)(21,24,23,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,15,23)(16,25,26,27)(20,28,22,24)Polycyclic presentation:
2^8|a^2=ef,b^2=h,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #273288, #273290, #273303, … |
| maximal normal subgroup of | #273288, #273290, #273303, … |
| maximal quotient of | #330577, #330578, #330579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22825);