Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13436 |
| automorphism | Aut(G) = #72317013 (order 4096) · Inn(G) = #2044 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
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 | |
219 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 219 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)(3,13)(4,8)(5,19)(6,10)(7,12)(9,16)(11,20)(14,27)(15,28)(17,29)(18,26)(21,23)(22,31)(24,30)(25,32);(2,8)(3,14)(4,6)(5,20)(10,12)(11,26)(15,17)(16,30)(19,23)(22,24)(27,28)(31,32);(1,3,4,15,7,17,6,14)(2,9,10,24,12,25,8,22)(5,16,18,31,20,32,21,30)(11,13,26,27,19,29,23,28)Polycyclic presentation:
2^8|c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,f^b=fh,g^c=gh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151240, #151241, #151264, … |
| maximal normal subgroup of | #151240, #151241, #151264, … |
| maximal quotient of | #144674, #144710, #144743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6424);