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 | #3169 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
322 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 322 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:
(1,2,5,8)(3,10,11,7)(4,6,13,9)(12,15,16,14)(17,18,21,25,22,19,26,24)(20,27,28,31,29,23,30,32);(2,6)(4,13)(7,14)(8,9)(10,15)(12,16)(17,19)(18,22)(20,27)(21,24)(23,29)(25,26)(28,31)(30,32);(1,3)(2,7)(4,12)(5,11)(6,14)(8,10)(9,15)(13,16)(17,20,26,30,22,29,21,28)(18,23,24,31,19,27,25,32);(2,8)(3,11)(6,9)(12,16)(18,19)(23,27)(24,25)(31,32)Polycyclic presentation:
2^8|a^2=f,c^2=fg,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 | #136569, #137497, #137499, … |
| maximal normal subgroup of | #136569, #137497, #137499, … |
| maximal quotient of | #182574, #183682, #183948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11079);