Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 131072 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,19,16,18);(2,3)(5,6,8,10)(7,9,12,11)(13,15,17,16)(14,18,20,19);(1,2,4,3);(1,3,4,2)(5,6)(7,9)(8,10)(11,12);(5,7,8,12)(6,9,10,11)(13,16,17,15)(14,19,20,18)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #98629, #251522, #251523, … |
| maximal normal subgroup of | #98629, #251522, #251523, … |
| maximal quotient of | #531683, #531684, #531776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52302);