Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,18,21)(15,16,23,19)(17,25,26,20)(22,27,28,24);(6,9)(10,12)(13,15,18,23)(14,19,21,16)(17,22,26,28)(20,27,25,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(5,6)(7,10)(8,9)(11,12)(13,16,18,19)(14,15,21,23)(17,24,26,27)(20,22,25,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,25)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97231, #131978, #132040, … |
| maximal normal subgroup of | #97231, #131978, #132040, … |
| maximal quotient of | #553733, #553996, #554189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29696);