Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27);(6,9)(10,12)(14,18)(15,22)(20,24)(23,28);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27);(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #6398553, #6398557, #6398564, … |
| maximal normal subgroup of | #6398553, #6398557, #6398564, … |
| maximal quotient of | #513830, #513849, #513863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27470);