Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
543 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(1,2,3,4)(6,9)(7,11)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,28)(26,27);(5,6)(7,10)(8,9)(11,12)(14,18)(19,27)(20,25)(21,28);(6,9)(10,12)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(5,7,8,11)(6,10,9,12)(14,18)(19,27)(20,25)(21,28)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107322, #107827, #112478, … |
| maximal normal subgroup of | #107322, #107827, #112478, … |
| maximal quotient of | #216342, #216348, #216372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14880);