Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13429 |
| automorphism | Aut(G) = #72317013 (order 4096) · Inn(G) = #2044 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2044 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3199 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
217 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,7)(5,10)(6,11)(8,15)(9,16)(13,25)(14,29)(17,21)(18,23)(19,26)(20,30)(22,32)(24,31)(27,28);(2,7)(3,13)(4,16)(8,20)(9,11)(10,21)(12,27)(17,31)(22,23)(24,26)(25,29)(30,32);(1,3,4,14,6,13,16,28)(2,8,9,22,11,23,7,20)(5,18,17,32,19,15,31,30)(10,25,24,29,26,12,21,27)Polycyclic presentation:
2^8|c^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,e^c=eh,f^c=fh,g^c=gh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151236, #151237, #151268, … |
| maximal normal subgroup of | #151236, #151237, #151268, … |
| maximal quotient of | #144670, #144706, #144738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6417);