Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1442 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,7,12,22)(4,14)(5,18)(6,9)(8,16)(10,19)(11,25,15,30)(13,29,28,21)(17,31)(20,32,23,27)(24,26);(1,3,5,13,6,12,19,28)(2,7,10,21,9,22,18,29)(4,15,16,32,17,11,26,27)(8,23,14,30,24,20,31,25);(1,4)(2,8,9,24)(3,11,12,15)(5,16)(6,17)(7,20)(10,14,18,31)(13,27,28,32)(19,26)(21,25)(22,23)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11273, #359161, #360496, … |
| maximal normal subgroup of | C2×#11273, #359161, #360496, … |
| maximal quotient of | #120666, #120718, #120720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4261);