Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 608 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) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,15)(16,24,27,21)(17,20,28,23)(18,25,26,22);(6,9)(10,12)(13,15,19,14)(16,23,27,20)(17,21,28,24)(18,25,26,22);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,26)(18,28)(19,27)(21,22)(24,25);(13,17,19,28)(14,21,15,24)(16,26,27,18)(20,22,23,25)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,d^b=dh,f^c=fh,g^c=gh,f^d=fh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #249918, #249922, #249933, … |
| maximal normal subgroup of | #249918, #249922, #249933, … |
| maximal quotient of | #349970, #349971, #350179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26007);