Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
551 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 191 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:
(6,9)(10,12)(13,14,17,15)(16,18,20,19)(21,22,25,26)(23,27,28,24);(1,2,3,4)(7,11)(10,12)(14,15)(16,20)(23,28)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,18,20,19)(21,23)(22,24)(25,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,20)(14,18,15,19)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #100782, #100786, #100789, … |
| maximal normal subgroup of | #100782, #100786, #100789, … |
| maximal quotient of | #154051, #154057, #154061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7656);