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) = C22×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 | |
635 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 635 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)(6,9)(7,11)(13,14)(15,18)(16,17)(19,20)(21,22,25,24)(23,27,28,26);(7,11)(10,12)(14,17)(18,19)(21,23,25,28)(22,26,24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,22)(23,26,28,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,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 | #17223⋊C3 |
| maximal subgroup of | C2×#17223, #845039, #845045, … |
| maximal normal subgroup of | C2×#17223, #845039, #845045, … |
| maximal quotient of | #156493, #156494, #156520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10211);