Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 327 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16)(8,21,24,35)(9,25)(10,14)(11,15)(12,18)(13,23,28,22)(17,31,34,26)(19,27)(20,30)(29,33,36,32);(1,2)(3,4)(5,7,9,18,11,20,10,19)(6,12,14,16,15,27,25,30)(8,17,22,32,24,34,23,33)(13,26,21,29,28,31,35,36);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,21,15,28,25,35)(7,17,18,32,20,34,19,33)(12,26,16,29,27,31,30,36)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131409, #131410, #144746, … |
| maximal normal subgroup of | #131409, #131410, #144746, … |
| maximal quotient of | #120165, #120179, #120180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4217);