Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,18,22,19,23,17,21)(15,20,16,25,24,27,26,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,18)(17,26)(19,24)(21,28)(22,25)(23,27);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,26)(14,20,23,27)(15,17,24,18)(21,25,22,28);(6,11)(10,12)(13,17,19,18)(14,21,23,22)(15,16,24,26)(20,25,27,28)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24615, #6453141, #6453146, … |
| maximal normal subgroup of | C2×#24615, #6453141, #6453146, … |
| maximal quotient of | #275552, #275554, #275560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17603);