Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,22)(16,18)(17,21)(19,23)(20,25)(24,28)(26,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(14,18)(16,25)(19,22)(20,21)(23,26)(27,28);(6,10)(7,12)(13,15,17,24)(14,19,21,27)(16,26,25,23)(18,22,20,28)Polycyclic presentation:
2^8|d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,f^c=fh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632014, #6632018, #6632026, … |
| maximal normal subgroup of | #6632014, #6632018, #6632026, … |
| maximal quotient of | #331656, #331657, #331660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23385);