Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1600 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
466 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)(4,15)(8,19)(9,18)(10,16)(13,23)(17,25)(20,26)(21,29)(22,30)(27,28)(31,32);(1,2,5,10)(3,8,11,20)(4,16,15,7)(6,9,12,21)(13,26,23,19)(14,22,24,32)(17,29,25,18)(27,30,28,31);(3,11)(4,17)(7,18)(8,20)(13,27)(14,24)(15,25)(16,29)(19,31)(22,32)(23,28)(26,30);(1,3,12,24)(2,8,21,32)(4,13,25,27)(5,11,6,14)(7,19,29,31)(9,22,10,20)(15,23,17,28)(16,26,18,30)Polycyclic presentation:
2^8|b^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18250, #6346537, #6346543, … |
| maximal normal subgroup of | C2×#18250, #6346537, #6346543, … |
| maximal quotient of | #166323, #166335, #166380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11238);