Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(9,10,12,11)(13,14,16,18)(15,19,20,17);(5,6,7,8)(9,11,12,10)(13,14)(15,17)(16,18)(19,20);(1,2,3,4)(5,6)(7,8)(13,14)(15,17)(16,18)(19,20);(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^8|a^2=h,b^2=egh,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #100124, #100140, #128418, … |
| maximal normal subgroup of | #100124, #100140, #128418, … |
| maximal quotient of | #153754, #153777, #153821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6991);