Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
693 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 693 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(2,4)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,25,24)(23,27,28,26);(6,9)(10,11)(15,19)(17,20)(21,23,25,28)(22,26,24,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,19)(18,20)(21,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^8|a^2=fg,b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #142555, C2×#15579, #609165, … |
| maximal normal subgroup of | #142555, C2×#15579, #609165, … |
| maximal quotient of | #154713, #154859, #154964, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8567);