Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #396 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1439 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
432 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)(3,11)(4,15,18,8)(5,19)(6,10)(7,13)(9,21)(12,28,16,26)(14,27)(17,24,20,25)(22,30,32,29)(23,31);(1,3,5,13,6,14,21,31)(2,7,9,11,10,23,19,27)(4,16,17,30,18,12,20,29)(8,22,24,28,15,32,25,26);(1,4)(2,8)(3,12)(5,20)(6,18)(7,22)(9,25)(10,15)(11,26)(13,30)(14,16)(17,21)(19,24)(23,32)(27,28)(29,31)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11138, #359387, #360321, … |
| maximal normal subgroup of | C2×#11138, #359387, #360321, … |
| maximal quotient of | #116993, #116994, #117025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4126);