Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(17,25)(21,28)(23,24)(26,27);(1,2)(3,4)(13,14,18,22)(15,20,16,19)(17,21,25,28)(23,27,24,26);(5,6,8,9)(7,10,11,12)(13,15,18,16)(14,19,22,20)(17,23,25,24)(21,26,28,27);(13,16,18,15)(14,20,22,19)(17,24,25,23)(21,27,28,26);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #111708, #111720, #111958, … |
| maximal normal subgroup of | #111708, #111720, #111958, … |
| maximal quotient of | #516498, #516563, #516657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28236);