Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) = C23×#612424 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18,21,25,22,24,30,31)(19,23,27,20,26,28,32,29);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,23)(20,25)(21,27)(22,26)(24,28)(29,31)(30,32);(17,20,21,28,22,29,30,23)(18,19,25,27,24,26,31,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(18,24)(19,26)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=f,c^2=fg,d^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dgh,e^a=eg,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #136754, #210397, #210399, … |
| maximal normal subgroup of | #136754, #210397, #210399, … |
| maximal quotient of | #182482, #182484, #183486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10974);