Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24);(5,6,7,8)(11,17)(15,18)(19,23)(22,24);(1,2)(3,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #36018⋊C3 |
| maximal subgroup of | #129788, #214805, #214821, … |
| maximal normal subgroup of | #129788, #214805, #214821, … |
| maximal quotient of | #517634, #517668, #517679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29006);