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) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
585 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22,25,24)(23,27,28,26);(6,8)(10,11)(15,19)(17,20)(21,23,25,28)(22,26,24,27);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^8|a^2=efg,b^2=ef,c^2=g,d^2=e,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 | #142590, C2×#15798, #609498, … |
| maximal normal subgroup of | #142590, C2×#15798, #609498, … |
| maximal quotient of | #154770, #154975, #155029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8786);