Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator | C2 | C22×C4×C8 |
| socle | C22 | C23×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,13,9,14,18,26,10,12,19,24,20,25,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,21)(13,22)(14,23)(18,29)(19,27)(20,28)(24,31)(25,32)(26,33)(30,35)(34,36);(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36);(6,12)(11,21)(13,24)(14,25)(22,31)(23,32)(26,34)(33,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,g^2=h,d^a=dh,c^b=ch136 conjugacy classes — view the full character table.
| maximal subgroup of | #126701, #137597, #137598, … |
| maximal normal subgroup of | #126701, #137597, #137598, … |
| maximal quotient of | #479897, #479909, #479915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26312);