Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33404 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#106 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,18)(4,14)(5,16)(8,17,20,28)(10,23,22,19)(12,26)(13,25)(15,27)(21,31,30,29)(24,32);(1,2,4,7,5,10,14,19,6,9,15,18,16,22,27,23)(3,8,12,17,13,21,26,29,11,20,24,28,25,30,32,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,17)(9,20)(10,21)(14,26)(15,24)(16,25)(18,28)(19,29)(22,30)(23,31)(27,32);(3,11)(8,20)(12,24)(13,25)(17,28)(21,30)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=e,e^2=g,g^2=h,b^a=be,e^a=eg,g^a=gh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #33404⋊C3 |
| maximal subgroup of | #497838, #497839, #498379, … |
| maximal normal subgroup of | #497838, #497839, #498379, … |
| maximal quotient of | #483326, #483399, #483425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26392);