Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | #3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10,14,17,15,11,13,19)(12,16,22,23,20,18,21,24);(1,2)(3,7)(4,6)(5,8)(9,11,15,10)(12,18,20,16)(13,17,14,19)(21,23,22,24);(9,12)(10,16)(11,18)(13,21)(14,22)(15,20)(17,23)(19,24);(12,20)(16,18)(21,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,f^2=g,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dg61 conjugacy classes — view the full character table.
| maximal subgroup of | #221306, #221307, #221310, … |
| maximal normal subgroup of | #221306, #221307, #221310, … |
| maximal quotient of | #160505, #160509, #160513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10641);