Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 262144 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(10,12)(13,14,18,22)(15,16,20,19)(17,25,26,21)(23,27,28,24);(1,2,3,4)(15,20)(16,19)(23,28)(24,27);(5,6,8,9)(7,10,11,12)(13,15,18,20)(14,19,22,16)(17,23,26,28)(21,27,25,24);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,28,25,23);(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=g,e^2=h,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 | #112606, #112608, #196299, … |
| maximal normal subgroup of | #112606, #112608, #196299, … |
| maximal quotient of | #516544, #516545, #516700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28314);