Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
505 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(2,4)(5,6,8,9)(7,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,22,26,25)(23,27,24,28);(6,9)(10,11)(14,16)(15,19)(21,23,26,24)(22,27,25,28);(5,7,8,12)(6,10,9,11)(13,15,17,19)(14,18,16,20)(21,24,26,23)(22,27,25,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,16,17,14)(15,18,19,20)(21,25,26,22)(23,27,24,28)Polycyclic presentation:
2^8|a^2=efg,b^2=f,c^2=ef,d^2=fg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| Fitting subgroup of | #17230⋊C3 |
| maximal subgroup of | #125788, #126105, #126197, … |
| maximal normal subgroup of | #125788, #126105, #126197, … |
| maximal quotient of | #156501, #156559, #156615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10218);