Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1551 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)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(5,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(1,2,3,4)(11,18)(14,17)(19,24)(21,23);(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=cf88 conjugacy classes — view the full character table.
| Frattini subgroup of | #72293085 |
| maximal subgroup of | #106018, #108000, #108058, … |
| maximal normal subgroup of | #106018, #108000, #108058, … |
| maximal quotient of | #513384, #513397, #513407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27058);