Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
254 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,11,15,10,25)(7,16,18,31,20,32,19,35)(8,21,22,27,24,36,23,13)(12,26,28,34,29,33,30,17);(5,7,8,17,9,18,22,26,11,20,24,34,10,19,23,33)(6,12,27,32,25,30,21,31,15,29,13,16,14,28,36,35);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #130202, #130203, #130252, … |
| maximal normal subgroup of | #130202, #130203, #130252, … |
| maximal quotient of | #137607, #137608, #137649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6051);