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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(2,6)(4,7)(10,13)(11,15)(18,19)(23,25)(24,26)(30,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,5,8)(2,7,6,4)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,4,5,7)(2,8,6,3)(9,11)(10,14)(12,15)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=fgh,e^2=gh,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×#40369, #7637716, #7638137, … |
| maximal normal subgroup of | C2×#40369, #7637716, #7638137, … |
| maximal quotient of | #522009, #522533, #523081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33357);