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) = C22×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 | |
649 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 179 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)(6,9)(7,11)(13,14,17,15)(16,19,20,18)(23,27)(26,28);(6,9)(10,12)(14,15)(16,20)(21,22,25,24)(23,26,27,28);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^8|a^2=e,b^2=efg,c^2=g,d^2=f,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 | #142573, C2×#15854, #609585, … |
| maximal normal subgroup of | #142573, C2×#15854, #609585, … |
| maximal quotient of | #154792, #154902, #154985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8842);