Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1997 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1997 conjugacy classes of subgroups — too many to draw. Only its 191 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,22)(19,21)(23,24);(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(2,4)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(6,8)(11,18)(15,22)(19,24)(21,23)Polycyclic presentation:
2^8|e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #32884⋊C3 |
| maximal subgroup of | #249879, #249891, #249896, … |
| maximal normal subgroup of | #249879, #249891, #249896, … |
| maximal quotient of | #346398, #346400, #346403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25872);