Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | Aut(G) = #72303733 (order 2048) · Inn(G) = #410 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #410 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | #1464 |
| prime radical | C24 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,4,10,5,11,17,27,6,9,16,25,18,8,22,24)(3,12,14,26,15,23,29,32,13,7,19,21,28,20,30,31);(1,3)(2,7)(4,14)(5,15)(6,13)(8,23)(9,12)(10,26)(11,20)(16,19)(17,29)(18,28)(21,25)(22,30)(24,31)(27,32);(2,8,9,11)(3,13)(4,16)(7,20,12,23)(10,27,25,24)(15,28)(17,22)(21,32,26,31)Polycyclic presentation:
2^8|a^2=e,c^2=f,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133265, #133266, #133275, … |
| maximal normal subgroup of | #133265, #133266, #133275, … |
| maximal quotient of | #124680, #124707, #124730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4510);