Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C22×#120 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2382 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)(3,12)(4,9)(5,11)(6,18)(7,10)(8,14)(13,28)(15,23)(16,25)(17,24)(19,22)(20,26)(21,30)(27,31)(29,32);(1,3)(2,8)(4,13)(5,14)(6,19)(7,15)(9,21)(10,22)(11,23)(12,18)(16,29)(17,31)(20,30)(24,32)(25,28)(26,27);(1,4)(2,9)(3,13)(5,16)(6,20)(7,17)(8,21)(10,24)(11,25)(12,27)(14,29)(15,31)(18,26)(19,30)(22,32)(23,28);(2,10)(8,22)(9,24)(11,18)(12,23)(21,32)(25,26)(27,28);(1,5,7,6)(2,11,10,18)(3,14,15,19)(4,16,17,20)(8,23,22,12)(9,25,24,26)(13,29,31,30)(21,28,32,27)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147757, #147758, #364521, … |
| maximal normal subgroup of | #147757, #147758, #364521, … |
| maximal quotient of | #6346377, #6346383, #6346389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53375);