Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
1008 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)(14,18)(15,21)(16,24)(26,28);(6,9)(10,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,11)(9,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28);(1,3,4,2)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=f,e^2=h,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 | C2×#34516, #7626149, #7626155, … |
| maximal normal subgroup of | C2×#34516, #7626149, #7626155, … |
| maximal quotient of | #513869, #513870, #513873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27504);