Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10004 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
267 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3,6,13)(2,7,10,15)(4,14,5,12)(8,16,9,11)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(20,30)(22,31)(23,28)(26,29)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10010, #235323, #235325, … |
| maximal normal subgroup of | C2×#10010, #235323, #235325, … |
| maximal quotient of | #102752, #102759, #102765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2998);