Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1065 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 1065 conjugacy classes of subgroups — too many to draw. Only its 559 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20);(6,10,13,12)(11,17,18,19);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(7,14)(11,18)(15,20)(17,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=be,e^a=eh,g^a=gh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #20343⋊C3 |
| maximal subgroup of | #356496, #356497, #356674, … |
| maximal normal subgroup of | #356496, #356497, #356674, … |
| maximal quotient of | #180896, #180899, #180902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13331);