Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #393 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1446 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,18,25)(3,7,13,21,12,20,17,26)(4,14,16,29,15,8,22,24)(11,23,28,32,27,19,30,31);(1,3)(2,7,9,20)(4,11)(5,17)(6,12)(8,23,14,19)(10,26,25,21)(13,18)(15,27)(16,30)(22,28)(24,31,29,32);(1,4)(2,8,9,14)(3,11)(5,16)(6,15)(7,19,20,23)(10,24,25,29)(12,27)(13,28)(17,30)(18,22)(21,31,26,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,g^b=gh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #11395⋊C3 |
| maximal subgroup of | C2×#11395, #359177, #363955, … |
| maximal normal subgroup of | C2×#11395, #359177, #363955, … |
| maximal quotient of | #121987, #122040, #122041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4383);