Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24);(5,6,8,7)(10,14)(15,17)(16,20)(22,23);(1,2,3,4)(5,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fgh,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #97683, #129344, #146708, … |
| maximal normal subgroup of | #97683, #129344, #146708, … |
| maximal quotient of | #513576, #513577, #513610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27264);