Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
603 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 603 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,19)(15,20,18,16)(21,22,25,27)(23,28,26,24);(6,9)(7,11)(15,18)(16,20)(21,23,25,26)(22,24,27,28);(5,7)(6,10)(8,11)(9,12)(13,15,17,18)(14,16,19,20)(21,23)(22,24)(25,26)(27,28);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,16)(14,18)(15,19)(17,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^8|a^2=g,b^2=efg,c^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| Fitting subgroup of | #17226⋊C3 |
| maximal subgroup of | C2×#17226, #845044, #845051, … |
| maximal normal subgroup of | C2×#17226, #845044, #845051, … |
| maximal quotient of | #156496, #156501, #156515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10214);