Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #388 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1443 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,8)(4,16)(5,20)(6,11)(7,10)(9,18)(13,30,32,26)(14,24,21,25)(17,23,31,29)(19,28)(22,27);(1,3,5,14,7,15,6,21)(2,8,20,25,10,12,11,24)(4,17,18,13,19,31,22,32)(9,26,28,23,27,30,16,29);(1,4)(2,9,10,27)(3,13)(5,18)(6,22)(7,19)(8,23,12,29)(11,16,20,28)(14,31)(15,32)(17,21)(24,26,25,30)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,e^2=h,f^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=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11313, #358422, #360366, … |
| maximal normal subgroup of | C2×#11313, #358422, #360366, … |
| maximal quotient of | #121082, #121107, #121108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4301);