Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(5,6,8,10)(7,11,12,9)(13,14,17,21,18,23,16,22)(15,24,25,20,27,28,26,19);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,25)(17,26)(18,27)(20,23)(21,28)(22,24);(2,4)(13,15)(14,20)(16,25)(17,26)(18,27)(19,23)(21,24)(22,28);(13,16,18,17)(14,21,23,22)(15,25,27,26)(19,28,20,24)Polycyclic presentation:
2^8|a^2=e,b^2=ef,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #390957, #390964, #391166, … |
| maximal normal subgroup of | #390957, #390964, #391166, … |
| maximal quotient of | #243833, #243850, #243867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16721);