Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2328 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 149 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 — 4 generators on 32 points:
(1,2)(3,6)(4,15,19,13)(5,20)(7,11)(8,12)(9,18,25,17)(10,21)(14,23)(16,29,32,24)(22,26)(27,30,28,31);(1,3,7,12,8,11,6,2)(4,9,17,15,19,25,18,13)(5,14,22,26,23,20,21,10)(16,24,30,28,32,29,31,27);(1,4)(2,9,12,25)(3,13,11,15)(5,16)(6,17)(7,18)(8,19)(10,24,26,29)(14,27,20,28)(21,30)(22,31)(23,32);(1,5,7,22,8,23,6,21)(2,10,3,14,12,26,11,20)(4,16,18,31,19,32,17,30)(9,24,13,27,25,29,15,28)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=ef,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #496130, #496132, #497148, … |
| maximal normal subgroup of | #496130, #496132, #497148, … |
| maximal quotient of | #219487, #219488, #219490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15274);