Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,11)(5,17)(9,19)(10,20)(12,25)(13,14)(16,28)(18,22)(24,30)(26,27)(29,32);(1,2,5,7,6,10,17,20)(3,8,13,18,14,23,11,22)(4,9,16,19,15,24,28,30)(12,21,27,29,26,31,25,32);(1,3)(2,8)(4,12)(5,13)(6,14)(7,18)(9,21)(10,23)(11,17)(15,26)(16,27)(19,29)(20,22)(24,31)(25,28)(30,32);(1,4)(2,9)(3,12)(5,16)(6,15)(7,19)(8,21)(10,24)(11,25)(13,27)(14,26)(17,28)(18,29)(20,30)(22,32)(23,31);(4,15)(9,24)(12,26)(16,28)(19,30)(21,31)(25,27)(29,32)Polycyclic presentation:
2^8|b^2=f,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60470⋊C3 |
| maximal subgroup of | #6574107, #6574111, #6574112, … |
| maximal normal subgroup of | #6574107, #6574111, #6574112, … |
| maximal quotient of | #6353803, #6353806, #6353811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53458);