Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C22×C22⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,20)(4,15)(8,19,23,30)(10,21,11,16)(13,27)(17,29)(22,28,24,31)(25,32);(1,2)(3,8)(4,16)(5,10)(6,11)(7,17)(9,18)(12,22)(13,28)(14,24)(15,20)(19,25)(21,29)(23,26)(27,30)(31,32);(1,3,5,14)(2,8,10,24)(4,13,15,27)(6,12,18,26)(7,19,21,31)(9,23,11,22)(16,28,20,30)(17,25,29,32);(3,12)(8,22)(13,25)(14,26)(19,28)(23,24)(27,32)(30,31)Polycyclic presentation:
2^8|a^2=f,c^2=g,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17604, #6343436, #6343443, … |
| maximal normal subgroup of | C2×#17604, #6343436, #6343443, … |
| maximal quotient of | #163728, #163729, #163730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10592);