Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #376 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 135 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)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,23,31)(16,27,30,35)(19,24)(20,25)(21,26)(22,29,33,36)(32,34);(5,7)(6,11,14,23)(8,16)(9,17)(10,18)(12,22,25,33)(13,15,26,28)(19,29)(20,30)(21,31)(24,27,34,35)(32,36);(2,4)(5,8,9,19,10,20,21,32)(6,12,13,24,14,25,26,34)(7,16,17,29,18,30,31,36)(11,22,15,27,23,33,28,35)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,b^a=bd,c^a=ce,g^a=gh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93549, #134460, #134490, … |
| maximal normal subgroup of | #93549, #134460, #134490, … |
| maximal quotient of | #108389, #108390, #108391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3787);