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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1587 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)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24);(2,4)(10,15)(11,18)(17,20)(19,24);(1,2)(3,4)(5,6,7,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23);(1,2)(3,4)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)Polycyclic presentation:
2^8|c^2=fg,d^2=f,e^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 | #497362, #497366, #6533540, … |
| maximal normal subgroup of | #497362, #497366, #6533540, … |
| maximal quotient of | #521356, #522090, #522601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30670);