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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 423 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) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,17,21,18,22,30,36)(15,19,25,33,24,32,40,43)(16,26,27,34,28,35,29,20)(23,37,39,42,38,44,41,31);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,33)(22,32)(26,37)(27,39)(28,38)(29,41)(30,40)(34,42)(35,44)(36,43);(2,4)(13,16,18,28)(14,20,22,34)(15,23,24,38)(17,29,30,27)(19,31,32,42)(21,35,36,26)(25,41,40,39)(33,44,43,37);(5,7,8,12)(6,10,9,11)(15,24)(19,32)(23,38)(25,40)(31,42)(33,43)(37,44)(39,41)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28391, #6463076, #6463084, … |
| maximal normal subgroup of | C2×#28391, #6463076, #6463084, … |
| maximal quotient of | #320737, #320739, #320745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21379);