Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,22,25,28,24,26,32,35)(23,29,31,34,30,36,33,27);(1,2)(3,4)(6,10)(11,17)(12,18)(19,20)(22,26)(27,34)(28,35)(29,36);(2,4)(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,23,24,30)(22,27,26,34)(25,33,32,31)(28,36,35,29);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(23,30)(27,34)(29,36)(31,33)Polycyclic presentation:
2^8|a^2=f,c^2=e,d^2=eh,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 | #122553, #122567, #122593, … |
| maximal normal subgroup of | #122553, #122567, #122593, … |
| maximal quotient of | #330738, #330743, #330747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22867);