Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 119 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,21)(20,23)(22,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(3,7)(6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24);(1,3,4,7)(2,6,5,8)(12,20)(15,23)(18,24)(21,22)Polycyclic presentation:
2^8|d^2=eh,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=de37 conjugacy classes — view the full character table.
| Fitting subgroup of | #29686⋊C3 |
| maximal subgroup of | #487301, #487304, #487305, … |
| maximal normal subgroup of | #487301, #487304, #487305, … |
| maximal quotient of | #328215, #328216, #328217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22674);