Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,26,22,25,30,32)(19,23,27,20,24,29,31,28);(2,5)(6,8)(17,19,21,27,22,24,30,31)(18,23,26,20,25,29,32,28);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,24)(19,25)(21,29)(22,28)(23,30)(26,31)(27,32);(9,11,12,15)(10,14,13,16)(18,25)(20,28)(23,29)(26,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21437, #6372525, #6372528, … |
| maximal normal subgroup of | C2×#21437, #6372525, #6372528, … |
| maximal quotient of | #353847, #354005, #354130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14425);