Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,17,21,18,22,16,19)(15,23,24,20,26,27,25,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(2,4)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(13,16,18,17)(14,19,22,21)(15,24,26,25)(20,27,28,23)Polycyclic presentation:
2^8|a^2=e,b^2=ef,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #379519, #381675, #382091, … |
| maximal normal subgroup of | #379519, #381675, #382091, … |
| maximal quotient of | #243837, #243839, #243856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16725);