Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 65536 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(15,23)(16,22)(17,20)(18,21)(19,24)(25,28)(26,27);(1,3,4,2)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,27)(22,28);(6,9)(10,12)(13,16,18,17)(14,20,21,22)(15,24,25,26)(19,23,27,28);(5,7,8,11)(6,10,9,12)(14,21)(19,27)(20,22)(23,28)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,d^2=fh,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #116702, #116732, #117331, … |
| maximal normal subgroup of | #116702, #116732, #117331, … |
| maximal quotient of | #207300, #207312, #207324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14354);