Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×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 | |
551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(14,17)(15,19)(22,25)(26,27);(6,9)(10,12)(14,17)(18,20)(21,22,24,25)(23,27,28,26);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=ef,d^2=eg,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#14482, #520840, #520855, … |
| maximal normal subgroup of | C2×#14482, #520840, #520855, … |
| maximal quotient of | #153968, #153986, #154000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7470);