Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,13,14)(11,16,15,12)(17,18)(19,24)(20,23)(21,25)(22,27)(26,31)(28,30)(29,32);(10,14)(11,15)(17,19,23,22)(18,24,20,27)(21,29,30,26)(25,32,28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,20,23,18)(19,24,22,27)(21,28,30,25)(26,31,29,32);(9,12)(10,15)(11,14)(13,16)(17,21,23,30)(18,25,20,28)(19,26,22,29)(24,31,27,32);(1,3,4,7)(2,6,5,8)(17,22,23,19)(18,24,20,27)(21,29,30,26)(25,31,28,32)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=h,d^2=fh,e^2=fg,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 | C2×#37264, #7634623, #7634630, … |
| maximal normal subgroup of | C2×#37264, #7634623, #7634630, … |
| maximal quotient of | #553880, #553885, #554107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30252);