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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 401 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,19,22,25)(17,20,23,26)(21,28,24,27);(6,9)(10,12)(14,18)(15,21)(17,23)(19,27)(22,24)(25,28);(2,4)(7,11)(10,12)(13,15)(14,19)(16,22)(17,24)(18,25)(20,27)(21,23)(26,28);(5,7,8,11)(6,10,9,12)(14,18)(15,22)(20,26)(21,24)Polycyclic presentation:
2^8|a^2=ef,d^2=eh,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 | #273279, #273281, #273296, … |
| maximal normal subgroup of | #273279, #273281, #273296, … |
| maximal quotient of | #330583, #330584, #330585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22826);