Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C22×#72217875 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| 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 | #3505 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8,11,25)(3,14)(5,18)(9,27,28,30)(10,24,23,29)(12,26,13,21)(15,31)(17,19);(1,2)(3,9)(4,10)(5,21)(6,12)(7,13)(8,15)(11,22)(14,24)(16,30)(17,29)(18,25)(19,28)(20,27)(23,32)(26,31);(1,3)(2,9)(4,18)(5,16)(6,14)(7,17)(8,23)(10,25)(11,28)(12,24)(13,29)(15,32)(19,22)(20,31)(21,30)(26,27);(1,4,6,16)(2,10,12,30)(3,15,14,31)(5,19,18,17)(7,20,22,32)(8,24,26,9)(11,23,13,27)(21,28,25,29)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh49 conjugacy classes — view the full character table.
| maximal subgroup of | #253704, #253710, #253716, … |
| maximal normal subgroup of | #253704, #253710, #253716, … |
| maximal quotient of | #160658, #160659, #160666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10611);