Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | #2535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 487 conjugacy classes of subgroups — too many to draw. Only its 127 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)(7,11)(8,9)(10,12)(13,14)(15,21)(16,19)(17,20)(18,24)(22,26)(23,28)(25,27);(6,9)(10,11)(13,15,22,23,16,24,17,25)(14,18,26,27,19,21,20,28);(2,4)(15,23)(17,22)(18,27)(20,26)(21,28)(24,25);(5,7,8,12)(6,10,9,11)(14,19)(18,21)(20,26)(27,28)Polycyclic presentation:
2^8|b^2=gh,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #264645, #264646, #264649, … |
| maximal normal subgroup of | #264645, #264646, #264649, … |
| maximal quotient of | #316790, #316791, #316792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21132);