Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| socle | C2 | C22×#99 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
618 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)(3,11)(4,8)(5,17,18,9)(6,10)(7,14,19,26)(12,23)(13,24)(15,28,29,21)(16,22)(20,27,30,32)(25,31);(1,3)(2,7)(4,13)(5,14)(6,12)(8,20)(9,11)(10,19)(15,27)(16,25)(17,23)(18,26)(21,24)(22,30)(28,31)(29,32);(3,12)(7,19)(11,23)(13,25)(14,26)(20,30)(24,31)(27,32);(1,4,6,16)(2,8,10,22)(3,13,12,25)(5,15,18,29)(7,20,19,30)(9,21,17,28)(11,24,23,31)(14,27,26,32)Polycyclic presentation:
2^8|a^2=f,d^2=h,b^a=be,e^a=eg,g^a=gh,c^b=ch,f^b=fg,f^e=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #146610, #146613, #146615, … |
| maximal normal subgroup of | #146610, #146613, #146615, … |
| maximal quotient of | #479996, #480039, #480093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26335);