Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(5,6,9,12)(7,11,8,10)(13,14,16,21,17,20,24,19)(15,18,22,27,23,26,28,25);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,15,17,23)(14,18,20,26)(16,22,24,28)(19,25,21,27);(2,4)(14,19)(16,24)(18,25)(20,21)(22,28)(26,27);(5,8,9,7)(6,11,12,10)(14,20)(18,26)(19,21)(25,27)Polycyclic presentation:
2^8|a^2=ef,b^2=egh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100917, #100918, #150012, … |
| maximal normal subgroup of | #100917, #100918, #150012, … |
| maximal quotient of | #323320, #323327, #323334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21505);