Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 364 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,30)(22,31)(23,27)(26,28)(29,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,23,22)(18,24,27,31)(20,29,21,28)(25,32,30,26);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,23,21)(18,25,27,30)(19,28,22,29)(24,26,31,32);(17,21,23,20)(18,26,27,32)(19,28,22,29)(24,25,31,30)Polycyclic presentation:
2^8|b^2=efh,c^2=eh,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,c^b=ch,g^d=gh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #33089⋊C3 |
| maximal subgroup of | #6471792, #6471798, #6471848, … |
| maximal normal subgroup of | #6471792, #6471798, #6471848, … |
| maximal quotient of | #392342, #392775, #392854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26077);