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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#553 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
572 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 139 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)(7,12)(8,10)(9,11)(13,14)(15,22)(16,20)(17,19)(18,24)(21,26)(23,28)(25,27);(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,24)(17,23)(19,27)(20,22)(21,28)(25,26);(2,4)(5,8,9,7)(6,10,11,12)(14,19)(15,23)(17,26)(20,21)(24,25)(27,28);(6,11)(10,12)(14,20)(18,22)(19,21)(27,28)Polycyclic presentation:
2^8|b^2=eh,c^2=eh,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 | #138792, #138797, #267410, … |
| maximal normal subgroup of | #138792, #138797, #267410, … |
| maximal quotient of | #256613, #256615, #256621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17080);