Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 3 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,22,18,20,16,21)(15,23,24,27,26,19,25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,28)(22,27);(2,4)(5,8,9,7)(6,10,11,12)(14,20)(15,24)(16,17)(19,27)(23,28)(25,26)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137428, #137429, #137430, … |
| maximal normal subgroup of | #137428, #137429, #137430, … |
| maximal quotient of | #121618, #121645, #121650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4353);