Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(10,13)(14,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31);(2,3)(4,8)(18,22)(20,29)(23,25)(26,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=f,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45837, #7702476, #7702507, … |
| maximal normal subgroup of | C2×#45837, #7702476, #7702507, … |
| maximal quotient of | #557139, #557196, #557420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38825);