Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
547 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 547 conjugacy classes of subgroups — too many to draw. Only its 195 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(13,14,16,17)(15,18,19,20)(21,22)(23,28)(24,25)(26,27);(6,9)(10,12)(14,17)(15,19)(21,23,26,24)(22,25,27,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24,26,23)(22,25,27,28);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,25)(22,23)(24,27)(26,28)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=eh,d^2=g,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #140039, C2×#14606, #521109, … |
| maximal normal subgroup of | #140039, C2×#14606, #521109, … |
| maximal quotient of | #153984, #153986, #154020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7594);