Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24);(2,4)(10,14)(15,17)(16,20)(22,23);(1,2)(3,4)(5,6,7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23);(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)Polycyclic presentation:
2^8|c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #497367, #497369, #6533543, … |
| maximal normal subgroup of | #497367, #497369, #6533543, … |
| maximal quotient of | #521358, #522093, #522603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30664);