Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,13)(9,15)(10,23)(11,14)(12,18)(17,30)(19,27)(20,29)(21,28)(22,33)(24,26)(25,32)(31,35)(34,36);(5,7)(6,12)(8,17)(9,18)(10,24)(11,19)(13,25)(14,26)(15,27)(16,23)(20,31)(21,32)(22,34)(28,35)(29,36)(30,33);(1,3,4,2)(5,8)(6,13)(7,17)(9,21)(10,22)(11,20)(12,25)(14,29)(15,28)(16,30)(18,32)(19,31)(23,33)(24,34)(26,36)(27,35);(6,14)(8,20)(12,26)(15,23)(16,27)(17,31)(21,22)(32,34);(5,9,11,10)(6,15,14,23)(7,18,19,24)(8,21,20,22)(12,27,26,16)(13,28,29,33)(17,32,31,34)(25,35,36,30)Polycyclic presentation:
2^8|a^2=g,c^2=g,e^2=h,f^2=h,b^a=bf,d^a=dh,f^a=fh,f^b=fh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #146730, #146740, #146754, … |
| maximal normal subgroup of | #146730, #146740, #146754, … |
| maximal quotient of | #6344524, #6344560, #6344594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53155);