Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
311 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,24,28,25,27,33,35)(23,29,31,36,30,26,32,34);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,26)(24,32)(25,30)(27,29)(28,36)(31,33)(34,35);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,30)(26,29)(31,32)(34,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,e^2=h,b^a=bd,d^a=dg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #106191, #107487, #107646, … |
| maximal normal subgroup of | #106191, #107487, #107646, … |
| maximal quotient of | #94925, #94929, #94937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 904);