Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14,17,21,18,15,19,23)(16,20,25,27,24,22,26,28);(5,6,8,9)(7,10,11,12)(13,15,17,23,18,14,19,21)(16,22,25,28,24,20,26,27);(2,4)(6,9)(10,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,7,8,11)(6,10,9,12)(16,24)(20,22)(25,26)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #209568, #209570, #209572, … |
| maximal normal subgroup of | #209568, #209570, #209572, … |
| maximal quotient of | #218458, #218459, #218460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15122);