Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 343 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 — 4 generators on 36 points:
(1,2,4,3)(5,6,9,15,11,14,10,16)(7,17,19,26,20,30,24,12)(8,13,22,27,21,29,23,28)(18,31,33,35,32,36,34,25);(5,7,10,19,11,20,9,24)(6,12,16,17,14,26,15,30)(8,18,23,33,21,32,22,34)(13,25,28,31,29,35,27,36);(1,3,4,2)(5,8,11,21)(6,13,14,29)(7,18,20,32)(9,22,10,23)(12,25,26,35)(15,27,16,28)(17,31,30,36)(19,33,24,34);(6,14)(8,21)(12,26)(15,16)(17,30)(18,32)(22,23)(33,34)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,d^a=dh,e^b=eh,f^b=fh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #146878, #146884, #146911, … |
| maximal normal subgroup of | #146878, #146884, #146911, … |
| maximal quotient of | #183575, #183576, #183577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13448);