Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,15,23,14,19,13,18)(11,16,12,22,21,17,20,24);(2,4)(5,6)(7,8)(9,11)(10,16)(12,15)(13,20)(14,21)(17,19)(18,24)(22,23);(6,8)(9,12,14,20)(10,17,19,16)(11,13,21,15)(18,24,23,22);(2,4)(9,13,14,15)(10,18,19,23)(11,12,21,20)(16,22,17,24)Polycyclic presentation:
2^8|a^2=fh,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24397, #6437747, #6437753, … |
| maximal normal subgroup of | C2×#24397, #6437747, #6437753, … |
| maximal quotient of | #268800, #268802, #268808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17385);