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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#550 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,10,8)(7,12,9,11)(13,14,17,15)(16,19,20,18);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15,17,14)(16,19,20,18);(1,2)(3,4)(5,6,10,8)(7,12,9,11)(13,16,17,20)(14,18,15,19);(5,8,10,6)(7,12,9,11);(5,9,10,7)(6,11,8,12)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,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 | #36005⋊C3 |
| maximal subgroup of | #132344, #248677, #248689, … |
| maximal normal subgroup of | #132344, #248677, #248689, … |
| maximal quotient of | #517817, #517835, #517844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28993);