Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,7,10,6,20)(3,8,15,23,13,24,14,22)(4,16,17,25,18,26,19,9)(12,27,30,21,28,31,29,32);(1,3,6,15,7,13,5,14)(2,8,11,22,10,24,20,23)(4,12,17,29,18,28,19,30)(9,21,26,31,25,32,16,27);(1,4)(2,9)(3,12)(5,19)(6,17)(7,18)(8,21)(10,25)(11,26)(13,28)(14,30)(15,29)(16,20)(22,31)(23,27)(24,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10787, #354322, #355561, … |
| maximal normal subgroup of | C2×#10787, #354322, #355561, … |
| maximal quotient of | #105847, #105848, #105852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3775);