Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×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 | |
308 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,27,28,24)(20,23,29,32)(21,30,31,25);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,20,22,28,31,29)(18,23,25,27,26,32,30,24);(2,6)(5,8)(11,15)(13,16)(19,28)(20,29)(23,32)(24,27);(17,20,21,28,22,29,31,19)(18,24,25,23,26,27,30,32)Polycyclic presentation:
2^8|a^2=fg,b^2=ef,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #504962, #505038, #505448, … |
| maximal normal subgroup of | #504962, #505038, #505448, … |
| maximal quotient of | #360474, #362680, #365310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20837);