Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,21)(16,22)(17,20)(18,19)(23,27)(24,28)(25,26);(6,10)(8,11)(14,19)(15,22)(20,26)(23,28);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,28)(26,27);(5,7,9,12)(6,11,10,8)(13,16)(14,15)(17,24)(18,21)(19,22)(20,23)(25,27)(26,28);(5,8,9,11)(6,12,10,7)(13,17,18,25)(14,20,19,26)(15,23,22,28)(16,24,21,27)Polycyclic presentation:
2^8|a^2=fg,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39873, #7637650, #7638118, … |
| maximal normal subgroup of | C2×#39873, #7637650, #7638118, … |
| maximal quotient of | #521914, #522474, #522997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32861);