Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
368 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(1,2)(3,4)(6,9,10,11)(13,14,18,20,17,16,24,25)(15,19,23,27,22,21,26,28);(5,6)(7,11)(8,10)(9,12)(13,15,17,22)(14,19,16,21)(18,26,24,23)(20,28,25,27);(2,4)(7,12)(9,11)(13,16)(14,17)(15,21)(18,20)(19,22)(23,27)(24,25)(26,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #152552, #152561, #153130, … |
| maximal normal subgroup of | #152552, #152561, #153130, … |
| maximal quotient of | #97001, #97006, #97175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1495);