Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,24,14,12,13,18)(11,16,21,19,22,23,20,17);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,23)(13,20)(14,22)(15,21)(17,18)(19,24);(9,12,14,10)(11,19,22,17)(13,18,15,24)(16,20,23,21);(2,6)(3,8)(9,11)(10,17)(12,19)(13,21)(14,22)(15,20)(16,18)(23,24)Polycyclic presentation:
2^8|a^2=fh,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6667762, #6667764, #6667768, … |
| maximal normal subgroup of | #6667762, #6667764, #6667768, … |
| maximal quotient of | #342303, #342304, #342305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24043);