Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
924 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 449 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 — 5 generators on 24 points:
(3,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21);(2,4)(3,7)(10,12)(14,15);(1,2,5,4)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19,20,24)(18,21,22,23);(1,3,5,7)(2,6,4,8)(9,12,13,10)(11,15,16,14);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,15)(13,16)Polycyclic presentation:
2^8|a^2=g,c^2=fgh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #46330⋊C3 |
| maximal subgroup of | #130315, #133593, #485966, … |
| maximal normal subgroup of | #130315, #133593, #485966, … |
| maximal quotient of | #557118, #557410, #557846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39318);