Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C23 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
851 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 851 conjugacy classes of subgroups — too many to draw. Only its 485 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 — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,g^c=gh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #61821⋊C3 |
| maximal subgroup of | #245767, #245770, #245791, … |
| maximal normal subgroup of | #245767, #245770, #245791, … |
| maximal quotient of | #6386239, #6387981, #6392755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54809);