Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2673 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 2673 conjugacy classes of subgroups — too many to draw. Only its 567 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(2,4)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(6,8)(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)Polycyclic presentation:
2^8|e^2=h,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| Fitting subgroup of | #34632⋊C3 |
| maximal subgroup of | #113288, #113529, #194670, … |
| maximal normal subgroup of | #113288, #113529, #194670, … |
| maximal quotient of | #515029, #515037, #515121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27620);