Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| 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 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
291 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 153 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(19,25)(22,28)(26,31)(29,32);(2,7)(6,15)(8,12)(9,16)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(18,23)(22,28)(24,30)(29,32)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=dh,e^a=eg,c^b=ch,e^b=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #137837, #209093, #209095, … |
| maximal normal subgroup of | #137837, #209093, #209095, … |
| maximal quotient of | #182514, #182550, #183469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10922);