Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14,19,24)(15,23,18,20)(16,17,22,21)(25,28,27,26);(5,6)(7,9)(8,10)(11,12)(13,15,19,18)(14,20,24,23)(16,27,22,25)(17,28,21,26);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,19,22)(14,21,24,17)(15,25,18,27)(20,28,23,26);(13,17,19,21)(14,22,24,16)(15,26,18,28)(20,27,23,25);(5,7,8,11)(6,9,10,12)(13,18,19,15)(14,23,24,20)(16,27,22,25)(17,28,21,26)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6439929, #6439933, #6439936, … |
| maximal normal subgroup of | #6439929, #6439933, #6439936, … |
| maximal quotient of | #516410, #516424, #516599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28232);