Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
282 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 282 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,3,5,11)(4,10,13,16,14,8,12,15)(17,18,22,25,21,24,30,32)(19,26,28,29,27,23,31,20);(1,3,7,2)(4,10,14,8)(5,9,6,11)(12,15,13,16)(17,19)(18,23)(20,25)(21,27)(22,28)(24,26)(29,32)(30,31);(1,4,6,13,7,14,5,12)(2,8,9,15,3,10,11,16)(17,20,22,26,21,29,30,23)(18,19,25,28,24,27,32,31);(2,3)(8,10)(9,11)(15,16)(19,27)(20,29)(23,26)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dg,e^a=eg,d^b=dh,e^b=eg,d^c=dgh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136570, #137498, #137500, … |
| maximal normal subgroup of | #136570, #137498, #137500, … |
| maximal quotient of | #182575, #183682, #183947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11078);