Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,21)(17,20)(18,22)(19,26)(24,27)(25,28);(6,9)(10,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,23)(21,27)(22,28);(2,4)(5,7)(6,10)(8,11)(9,12)(14,20)(15,24)(16,18)(19,27)(23,28)(25,26);(7,11)(10,12)(13,16,17,18)(14,21,20,22)(15,25,26,24)(19,27,23,28)Polycyclic presentation:
2^8|a^2=fh,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29720, #6432971, #6432984, … |
| maximal normal subgroup of | C2×#29720, #6432971, #6432984, … |
| maximal quotient of | #329439, #329440, #329441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22708);