Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,32)(27,28)(29,31);(1,3)(2,6)(4,8)(5,7)(10,13)(11,15)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(3,8)(5,6)(9,11,12,15)(10,14,13,16)(18,23)(19,26)(24,31)(25,27)Polycyclic presentation:
2^8|a^2=efh,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #32867⋊C3 |
| maximal subgroup of | #123193, #123203, C2×#32867, … |
| maximal normal subgroup of | #123193, #123203, C2×#32867, … |
| maximal quotient of | #391449, #391474, #391496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25855);