Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
471 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)(7,11)(8,10)(9,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(1,2,3,4)(5,7,8,12)(6,9,10,11)(15,22)(18,27)(21,28)(23,26);(2,4)(13,15,16,23,17,22,25,26)(14,18,19,21,20,27,24,28);(6,10)(9,11)Polycyclic presentation:
2^8|a^2=h,b^2=egh,c^2=f,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28576, #6459534, #6459552, … |
| maximal normal subgroup of | C2×#28576, #6459534, #6459552, … |
| maximal quotient of | #323881, #323882, #323883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21564);