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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,13)(12,15)(14,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30);(2,5)(3,7)(9,11,12,16)(10,13,15,14)(18,23)(19,26)(24,30)(27,32);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31);(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)Polycyclic presentation:
2^8|b^2=fg,c^2=gh,e^2=f,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×#38115, #7635654, #7635667, … |
| maximal normal subgroup of | C2×#38115, #7635654, #7635667, … |
| maximal quotient of | #521705, #522322, #522825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31103);