Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C22×C22⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3519 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,20)(3,12)(4,15)(8,22,24,31)(10,21,11,16)(13,26)(14,29)(17,30)(19,23,32,25)(27,28);(1,2)(3,8)(4,16)(5,10)(6,11)(7,17)(9,18)(12,25)(13,23)(14,22)(15,20)(19,27)(21,30)(24,26)(28,32)(29,31);(1,3)(2,8)(4,14)(5,13)(6,12)(7,19)(9,24)(10,23)(11,25)(15,28)(16,22)(17,27)(18,26)(20,32)(21,31)(29,30);(3,13)(8,23)(12,26)(14,28)(19,31)(22,32)(24,25)(27,29)Polycyclic presentation:
2^8|a^2=f,e^2=h,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17597, #6343388, #6343389, … |
| maximal normal subgroup of | C2×#17597, #6343388, #6343389, … |
| maximal quotient of | #160630, #160637, #160642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10585);