Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#538 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,9,6,10,15,23)(3,8,13,21,11,22,25,32)(5,7,14,18,16,19,28,30)(12,17,26,29,24,20,27,31);(2,7)(3,11)(4,14)(8,20)(10,19)(12,24)(13,27)(15,28)(17,22)(21,32)(25,26)(29,31);(1,3)(2,8)(4,13)(5,12)(6,11)(7,17)(9,21)(10,22)(14,26)(15,25)(16,24)(18,29)(19,20)(23,32)(27,28)(30,31);(3,12)(8,17)(11,24)(13,26)(20,22)(21,29)(25,27)(31,32)Polycyclic presentation:
2^8|a^2=f,f^2=h,b^a=be,e^a=eg,c^b=ch,f^b=fg,d^c=dg58 conjugacy classes — view the full character table.
| maximal subgroup of | #213752, #213753, #213756, … |
| maximal normal subgroup of | #213752, #213753, #213756, … |
| maximal quotient of | #161570, #161576, #161582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10374);