Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 4096 |
| 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 | #466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
288 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(1,2,7,12)(3,13,4,9)(5,11,6,10)(8,15,16,14)(17,18,21,25,22,24,30,31)(19,23,27,20,26,28,32,29);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(17,19)(18,23)(20,25)(21,27)(22,26)(24,28)(29,31)(30,32);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,20,21,28,22,29,30,23)(18,19,25,27,24,26,31,32);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(18,24)(19,26)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dgh,e^a=eg,d^b=dh,e^b=eg,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #136318, #137469, #137470, … |
| maximal normal subgroup of | #136318, #137469, #137470, … |
| maximal quotient of | #182486, #183645, #183913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10981);