Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,11)(15,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31);(1,2,5,4)(3,6,7,8)(9,10,14,11)(12,15,13,16)(18,19)(21,29)(23,25)(27,32);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,16,13,15)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(9,12,14,13)(10,15,11,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,2)(3,8,7,6)(9,13,14,12)(10,16,11,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #35510⋊C3 |
| maximal subgroup of | #132126, #253325, #253347, … |
| maximal normal subgroup of | #132126, #253325, #253347, … |
| maximal quotient of | #515500, #515656, #516230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28498);