Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1115 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1115 conjugacy classes of subgroups — too many to draw. Only its 503 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 — 5 generators on 28 points:
(2,4)(7,11)(9,12)(13,14,18,21)(15,16,19,24)(17,25,26,20)(22,28,27,23);(1,2,3,4)(5,6,8,10)(7,9,11,12)(13,15)(14,16)(17,22)(18,19)(20,23)(21,24)(25,28)(26,27);(5,7,8,11)(6,9,10,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,25)(26,28);(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,25)(26,28);(13,17,18,26)(14,20,21,25)(15,22,19,27)(16,23,24,28)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #112021, #112032, #112056, … |
| maximal normal subgroup of | #112021, #112032, #112056, … |
| maximal quotient of | #516428, #516461, #516585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28308);