Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | Aut(G) = #72256403 (order 2048) · Inn(G) = #1790 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | #3545 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
379 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,7,12,5,11)(3,13,15,31,16,27,21,30)(4,10,19,28,20,29,18,26)(8,22,24,14,25,32,23,17);(1,3)(2,8)(4,17)(5,21)(6,15)(7,16)(9,23)(10,27)(11,24)(12,25)(13,29)(14,20)(18,32)(19,22)(26,30)(28,31);(2,9)(5,6)(8,23)(10,28)(11,12)(13,30)(15,21)(18,19)(22,32)(24,25)(26,29)(27,31)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eg,f^b=fgh,d^c=dgh,e^c=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152568, #152592, #152593, … |
| maximal normal subgroup of | #152568, #152592, #152593, … |
| maximal quotient of | #129585, #129586, #129588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5769);