Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1163 conjugacy classes of subgroups; only the 427 normal subgroups are shown.
This group has 1163 conjugacy classes of subgroups — too many to draw. Only its 427 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) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,15)(14,16);(2,3)(6,7)(10,11)(14,15)(19,23)(22,24);(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,23,24);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,20,24);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^8|a^2=fg,c^2=fgh,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53036);