Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C22×C4×D4 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11,7,3,12,10)(4,8,16,22,17,13,25,21)(5,9,19,24,18,14,26,23)(15,20,29,31,28,27,32,30);(1,3)(2,7)(4,13)(5,14)(6,10)(8,17)(9,18)(11,12)(15,27)(16,21)(19,23)(20,28)(22,25)(24,26)(29,30)(31,32);(1,4,6,16,7,17,12,25)(2,8,11,22,3,13,10,21)(5,15,19,29,18,28,26,32)(9,20,24,31,14,27,23,30);(1,5)(2,9)(3,14)(4,15)(6,19)(7,18)(8,20)(10,23)(11,24)(12,26)(13,27)(16,29)(17,28)(21,30)(22,31)(25,32);(5,18)(9,14)(15,28)(19,26)(20,27)(23,24)(29,32)(30,31)Polycyclic presentation:
2^8|a^2=g,c^2=g,f^2=h,g^2=h,b^a=bf,f^a=fh,f^b=fh,e^d=eh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60183⋊C3 |
| maximal subgroup of | #129525, #129526, #481070, … |
| maximal normal subgroup of | #129525, #129526, #481070, … |
| maximal quotient of | C8×#576, C8×#577, C8×#578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53171);