Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7313 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1304 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C23×Q8 | C22 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,7,12,20,28)(3,13,15,29,16,23,31,32)(4,11,18,27,6,9,19,25)(8,21,22,14,24,30,26,17);(1,3,6,14,7,16,4,17)(2,8,11,23,12,24,9,13)(5,15,18,21,20,31,19,30)(10,26,25,32,28,22,27,29)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=h,e^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gh,c^b=cg,d^b=dfh,f^b=fh,d^c=dh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94632, #94633, C2×#7315, … |
| maximal normal subgroup of | #94632, #94633, C2×#7315, … |
| maximal quotient of | #93112, #93114, #93116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 303);