Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3489 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,14,16,19)(15,21,22,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,19)(13,16)(14,20)(17,22)(18,24)(21,23);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(11,17)(15,22)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #220530, #220531, #220536, … |
| maximal normal subgroup of | #220530, #220531, #220536, … |
| maximal quotient of | #163727, #163734, #163735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10596);