Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,24,27,25,28,33,36)(23,29,31,34,30,35,32,26);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(23,30)(26,34)(29,35)(31,32);(2,4)(8,16)(11,18)(14,20)(17,19)(21,23,25,30)(22,26,28,34)(24,32,33,31)(27,35,36,29);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)Polycyclic presentation:
2^8|a^2=ef,b^2=egh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #294800, #294802, #294823, … |
| maximal normal subgroup of | #294800, #294802, #294823, … |
| maximal quotient of | #331517, #331518, #331522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23269);