Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,16)(17,18)(19,20)(22,25)(23,27);(7,11)(9,12)(14,18)(15,19)(21,22,24,25)(23,26,27,28);(5,7)(6,9)(8,11)(10,12)(13,15,17,19)(14,16,18,20)(21,22)(23,28)(24,25)(26,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^8|a^2=eg,b^2=f,c^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16404, #610924, #610968, … |
| maximal normal subgroup of | C2×#16404, #610924, #610968, … |
| maximal quotient of | #155637, #155690, #155807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9392);