Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
557 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,17,14,18,12,15)(11,16,21,24,19,23,20,22);(2,5)(6,7)(10,15,18,17)(16,22,23,24);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,24)(18,23);(11,19)(16,23)(20,21)(22,24)Polycyclic presentation:
2^8|a^2=g,b^2=efg,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124683, #124690, #251672, … |
| maximal normal subgroup of | #124683, #124690, #251672, … |
| maximal quotient of | #206782, #206793, #206804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14318);