Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,18,20)(15,23,25,21)(16,19,26,28)(17,27,24,22);(1,2)(3,4)(13,15,24,26,18,25,17,16)(14,19,27,21,20,28,22,23);(2,4)(5,7,8,12)(6,9,10,11)(14,20)(19,28)(21,23)(22,27);(13,16,17,25,18,26,24,15)(14,21,22,19,20,23,27,28)Polycyclic presentation:
2^8|a^2=f,b^2=eh,c^2=fh,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #503657, #503693, #504126, … |
| maximal normal subgroup of | #503657, #503693, #504126, … |
| maximal quotient of | #305772, #305780, #305788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19051);