Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,9)(7,12,8,11)(13,14,18,21)(15,19,22,26)(16,17,20,25)(23,24,27,28);(5,7,10,8)(6,11,9,12)(15,22)(19,26)(23,27)(24,28);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28);(5,6,10,9)(7,11,8,12)(13,16)(14,17)(15,23)(18,20)(19,24)(21,25)(22,27)(26,28);(5,9,10,6)(7,12,8,11)(13,17)(14,20)(15,24)(16,21)(18,25)(19,27)(22,28)(23,26)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129904, #214798, #214846, … |
| maximal normal subgroup of | #129904, #214798, #214846, … |
| maximal quotient of | #517721, #517722, #517745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29021);