Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(10,14)(11,12)(20,27)(24,29)(26,31)(30,32);(2,3)(6,7)(9,10,13,14)(11,15,12,16)(18,22)(19,25)(24,29)(26,31);(9,11,13,12)(10,15,14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|b^2=g,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132128, #253367, #253389, … |
| maximal normal subgroup of | #132128, #253367, #253389, … |
| maximal quotient of | #553841, #554078, #554224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30015);