Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
305 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,13)(7,14,16,19)(8,12,9,17)(11,18,20,15)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,35)(34,36);(5,7,10,16)(6,11,13,20)(8,15,9,18)(12,19,17,14)(21,23,25,30)(22,26,28,33)(24,29,31,36)(27,34,35,32);(2,4)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,36)(33,34)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9929, #234468, #234472, … |
| maximal normal subgroup of | C2×#9929, #234468, #234472, … |
| maximal quotient of | #102938, #102941, #102943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2917);