Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 131 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)(15,23)(16,20)(17,21)(18,22)(19,26)(24,27)(25,28);(6,9)(10,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,25)(23,27)(26,28);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,27)(14,19,22,28)(16,25,17,26)(20,23,21,24);(7,11)(10,12)(13,16,18,17)(14,20,22,21)(15,25,27,26)(19,23,28,24)Polycyclic presentation:
2^8|c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=de,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #166747, #166749, #166779, … |
| maximal normal subgroup of | #166747, #166749, #166779, … |
| maximal quotient of | #326690, #326698, #326705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22617);