Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
547 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 547 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(13,14,17,15)(16,18,20,19)(22,23)(26,27);(6,9)(10,12)(14,15)(18,19)(21,22,25,23)(24,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,18,20,19)(21,23,25,22)(24,26,28,27);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=e,d^2=fgh,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| Fitting subgroup of | #14333⋊C3 |
| maximal subgroup of | C2×#14333, #515177, #515179, … |
| maximal normal subgroup of | C2×#14333, #515177, #515179, … |
| maximal quotient of | #153928, #153929, #153952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7321);