Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,25)(17,21)(18,26)(19,22)(23,27)(24,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,17)(18,23)(19,24)(21,25)(22,28)(26,27);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,20)(15,17)(18,24)(19,23)(21,27)(22,28)(25,26);(6,11)(10,12)(13,17,19,18)(14,21,22,26)(15,23,24,16)(20,27,28,25)Polycyclic presentation:
2^8|b^2=e,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24070, #6452898, #6452904, … |
| maximal normal subgroup of | C2×#24070, #6452898, #6452904, … |
| maximal quotient of | #256413, #256415, #256416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17058);