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) = C23×C4 |
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 127 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,25)(17,27)(18,22)(19,23)(21,26)(24,28);(1,3,4,2)(5,7,8,11)(6,9,10,12)(13,15,17,23,18,24,26,16)(14,19,21,20,22,25,27,28);(2,3)(6,10)(9,12);(13,16,26,24,18,23,17,15)(14,20,27,19,22,28,21,25)Polycyclic presentation:
2^8|a^2=g,b^2=ef,d^2=eh,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 | #504561, #504565, #504568, … |
| maximal normal subgroup of | #504561, #504565, #504568, … |
| maximal quotient of | #315015, #315021, #315027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20777);