Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 133 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)(7,12)(8,10)(9,11)(13,14,17,21)(15,23,26,19)(16,22,18,20)(24,28,25,27);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,27)(21,23)(22,28);(2,4)(5,8,9,7)(6,10,11,12)(14,20)(15,24)(16,18)(21,22)(25,26)(27,28);(6,11)(10,12)(13,16,17,18)(14,20,21,22)(15,25,26,24)(19,27,23,28)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=eh,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139221, #139289, #267958, … |
| maximal normal subgroup of | #139221, #139289, #267958, … |
| maximal quotient of | #256713, #256715, #256717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17099);