Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(1,3,4,5)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(2,6)(3,5)(7,8)(11,18)(14,17)(19,24)(21,23);(2,7)(6,8)(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^8|b^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #113129, #113154, #113225, … |
| maximal normal subgroup of | #113129, #113154, #113225, … |
| maximal quotient of | #6374646, #6374648, #6375031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53934);