Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(2,4)(5,6,9,14,27,15,25,31,11,17,24,13,10,16,8,23)(7,18,19,33,20,28,35,30,22,34,36,29,21,32,26,12);(1,2,3,4)(5,7)(6,12)(8,19)(9,26)(10,20)(11,22)(13,28)(14,32)(15,29)(16,33)(17,30)(18,23)(21,27)(24,35)(25,36)(31,34);(5,8,10,24,11,25,27,9)(6,13,15,23,17,14,16,31)(7,19,20,35,22,36,21,26)(12,28,29,18,30,32,33,34)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13260, #495271, #495272, … |
| maximal normal subgroup of | C2×#13260, #495271, #495272, … |
| maximal quotient of | #140848, #140870, #141022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6248);