Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,15)(16,24,27,28)(17,22,18,21)(20,26,23,25);(6,9)(10,11)(13,15,19,14)(16,24,27,28)(17,21,18,22)(20,26,23,25);(2,4)(13,16,19,27)(14,20,15,23)(17,26,18,25)(21,28,22,24);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #244882, #244884, #244886, … |
| maximal normal subgroup of | #244882, #244884, #244886, … |
| maximal quotient of | #258874, #258885, #258896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17179);