Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,22)(17,21)(18,23)(20,24)(25,28)(26,27);(1,3,4,2)(6,9)(10,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,21)(22,27)(23,28);(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,21)(22,28)(23,27);(13,16,18,17)(14,21,23,22)(15,24,25,26)(19,27,28,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #360877, #360878, #360879, … |
| maximal normal subgroup of | #360877, #360878, #360879, … |
| maximal quotient of | #202191, #202203, #202215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14027);