Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,17,25,18,16,19,21)(15,22,26,28,24,20,23,27);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,15,18,24)(14,20,16,22)(17,26,19,23)(21,28,25,27);(2,4)(13,16,18,14)(15,20,24,22)(17,25,19,21)(23,27,26,28);(5,8,9,7)(6,11,12,10)(13,17,18,19)(14,21,16,25)(15,23,24,26)(20,27,22,28)Polycyclic presentation:
2^8|a^2=ef,b^2=egh,c^2=h,d^2=e,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×#28526, #6466117, #6466126, … |
| maximal normal subgroup of | C2×#28526, #6466117, #6466126, … |
| maximal quotient of | #323602, #323604, #323610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21514);