Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
239 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 71 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 — 3 generators on 32 points:
(1,2,3,6,4,5,7,8)(9,10,12,14,13,16,11,15)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31);(10,14)(11,12)(15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(2,5)(6,8)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=g,e^2=h,b^a=bd,c^a=cfgh,d^a=df,d^b=dg,e^b=efg52 conjugacy classes — view the full character table.
| maximal subgroup of | #97990, #97993, #97995, … |
| maximal normal subgroup of | #97990, #97993, #97995, … |
| maximal quotient of | #96262, #96741, #96743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 630);