Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(5,6,10,15,11,14,9,24)(7,16,18,27,19,12,25,28)(8,20,22,29,23,35,21,13)(17,31,32,30,33,34,36,26);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,25)(11,19)(13,26)(14,16)(15,27)(20,34)(21,32)(22,36)(23,33)(24,28)(29,30)(31,35);(5,8,9,21,11,23,10,22)(6,13,15,20,14,29,24,35)(7,17,18,32,19,33,25,36)(12,26,27,34,16,30,28,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93576, #93582, #134423, … |
| maximal normal subgroup of | #93576, #93582, #134423, … |
| maximal quotient of | #108859, #108860, #108861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3855);