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) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,25,27,23)(20,22,29,30)(26,32,31,28);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,31)(30,32);(11,15)(13,16)(17,20,22,24,21,29,30,18)(19,26,28,25,27,31,32,23);(1,3,4,8)(2,6,5,7)(18,24)(19,27)(20,29)(28,32)Polycyclic presentation:
2^8|a^2=eg,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27904, #6458735, #6458741, … |
| maximal normal subgroup of | C2×#27904, #6458735, #6458741, … |
| maximal quotient of | #360673, #363352, #367289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20892);