Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| 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×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(10,12)(13,14,18,21)(15,17,23,19)(16,24,26,20)(22,27,28,25);(6,9)(7,11)(13,15,18,23)(14,19,21,17)(16,22,26,28)(20,27,24,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,21)(18,23)(20,27)(24,25)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,26)(14,20,21,24)(15,22,23,28)(17,25,19,27);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,15,21,23)(16,25,26,27)(20,22,24,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133405, #499801, #500239, … |
| maximal normal subgroup of | #133405, #499801, #500239, … |
| maximal quotient of | #557022, #557354, #557560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38441);