Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2699 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 2699 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:
(2,4)(6,8)(9,10,13,14)(11,15,19,17)(12,16,20,21)(18,22,24,23);(5,6)(7,8)(10,14)(12,20)(15,17)(18,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #35446⋊C3, #72241276, #72241277 |
| maximal subgroup of | #97556, #132081, #133718, … |
| maximal normal subgroup of | #97556, #132081, #133718, … |
| maximal quotient of | #515286, #515521, #515669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28434);