Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,14)(10,12)(11,16)(13,21)(17,33)(18,29)(20,28)(22,27)(23,26)(24,32)(25,34)(30,35)(31,36);(5,7,9,17,10,18,23,35)(6,11,14,25,12,24,26,36)(8,16,20,31,21,32,22,34)(13,15,27,30,19,29,28,33);(1,3,4,2)(6,12)(11,24)(13,19)(14,26)(15,29)(25,36)(27,28)(30,33)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,f^2=h,b^a=bd,c^a=ch,d^a=dh,f^a=fgh,d^b=dg,g^b=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130487, #130489, #130527, … |
| maximal normal subgroup of | #130487, #130489, #130527, … |
| maximal quotient of | #128312, #128338, #128370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5619);