Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 32768 |
| 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×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,16,20,18,22,17,21)(15,23,25,28,24,27,26,19);(1,2)(3,4)(6,9)(10,12)(13,15,18,24)(14,19,22,28)(16,26,17,25)(20,27,21,23);(2,4)(14,20)(16,17)(19,27)(21,22)(23,28)(25,26);(5,7,8,11)(6,10,9,12)(15,24)(19,28)(23,27)(25,26)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=ef,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 | #219491, #219509, #219510, … |
| maximal normal subgroup of | #219491, #219509, #219510, … |
| maximal quotient of | #244132, #244134, #244190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16797);