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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
812 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)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30);(2,5)(6,8)(10,13)(11,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,3,4,7)(2,6,5,8)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fgh,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×#38591, #7636207, #7636315, … |
| maximal normal subgroup of | C2×#38591, #7636207, #7636315, … |
| maximal quotient of | #521900, #522459, #523190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31579);