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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 465 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)(7,11)(8,9)(10,12)(13,14,19,25,18,16,17,21)(15,20,26,28,23,22,24,27);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,22)(17,24)(18,23)(19,26)(21,27)(25,28);(2,4)(13,16,18,14)(15,22,23,20)(17,21,19,25)(24,27,26,28);(5,7,8,12)(6,10,9,11)(15,23)(20,22)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=fh,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×#28392, #6463076, #6463084, … |
| maximal normal subgroup of | C2×#28392, #6463076, #6463084, … |
| maximal quotient of | #320801, #320803, #320809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21380);