Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
557 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(10,15)(11,19)(12,13)(17,18)(20,21)(22,23);(2,6)(3,4)(7,8)(9,10,14,18)(11,16,20,24)(12,15,13,17)(19,22,21,23);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,23,24,22)Polycyclic presentation:
2^8|b^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #249801, #249804, #249815, … |
| maximal normal subgroup of | #249801, #249804, #249815, … |
| maximal quotient of | #168925, #168982, #169025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11814);