Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,10,7,9,11,8,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(5,7,10,11,9,12,8,6)(13,15)(14,19)(16,25)(17,23)(18,24)(20,28)(21,22)(26,27);(1,2,3,4)(6,11)(7,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,22,28);(5,8,9,10)(6,7,11,12)(13,17,18,16)(14,20,21,26)(15,23,24,25)(19,28,22,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100652, #100656, #100660, … |
| maximal normal subgroup of | #100652, #100656, #100660, … |
| maximal quotient of | #219320, #219321, #219322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15232);