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 | #2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28);(6,10)(8,11)(14,18)(19,25)(20,23)(26,27);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27);(5,7,9,12)(6,11,10,8)(13,16)(14,20)(15,21)(17,24)(18,23)(19,26)(22,28)(25,27);(5,8,9,11)(6,12,10,7)(13,14)(15,19)(16,20)(17,18)(21,26)(22,25)(23,24)(27,28)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=h,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×#39870, #7637551, #7638118, … |
| maximal normal subgroup of | C2×#39870, #7637551, #7638118, … |
| maximal quotient of | #521908, #522468, #522989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32858);