Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,17)(16,18)(19,20);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,22)(23,26)(24,25)(27,28);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(22,25)(26,28);(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=e,d^2=g,b^a=be,c^a=cf,c^b=cg88 conjugacy classes — view the full character table.
| Fitting subgroup of | #13855⋊C3 |
| maximal subgroup of | #99140, #99141, #99142, … |
| maximal normal subgroup of | #99140, #99141, #99142, … |
| maximal quotient of | #153738, C4×#1443, #154127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6843);