Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 207 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)(6,9)(10,11)(13,14)(15,22)(16,20)(17,26)(18,21)(19,25)(23,28)(24,27);(5,6,8,9)(7,11,12,10)(15,23)(19,27)(22,28)(24,25);(13,15,17,25,18,23,16,24)(14,19,20,22,21,27,26,28);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,22,27,28)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #136381, #136441, #136922, … |
| maximal normal subgroup of | #136381, #136441, #136922, … |
| maximal quotient of | #226368, #226380, #226390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15718);