Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 526 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) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,19,24,32)(16,26,29,41)(17,30,27,21)(20,34,35,28)(23,36,38,44)(25,40,37,33)(31,42,43,39);(1,2)(3,4)(6,9)(10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,33)(22,32)(26,36)(27,37)(28,39)(29,38)(30,40)(34,42)(35,43)(41,44);(2,4)(13,16,27,34,18,29,17,28)(14,20,30,41,22,35,21,26)(15,23,37,42,24,38,25,39)(19,31,40,44,32,43,33,36);(5,7,8,12)(6,10,9,11)(15,24)(19,32)(23,38)(25,37)(31,43)(33,40)(36,44)(39,42)Polycyclic presentation:
2^8|a^2=h,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139471, #347169, C2×#25609, … |
| maximal normal subgroup of | #139471, #347169, C2×#25609, … |
| maximal quotient of | #293678, #293683, #293688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18597);