Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 163 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)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22)Polycyclic presentation:
2^8|a^2=f,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #99719, #99722, #150209, … |
| maximal normal subgroup of | #99719, #99722, #150209, … |
| maximal quotient of | #242781, #242782, #242783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16615);