Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,24);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18253, #6346558, #6346584, … |
| maximal normal subgroup of | C2×#18253, #6346558, #6346584, … |
| maximal quotient of | #166994, #167014, #167058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11241);