Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,7)(10,11)(13,14,18,22)(15,19,23,25)(16,20,26,28)(17,24,27,21);(2,4)(5,6,9,7)(8,11,12,10)(15,23)(17,27)(19,25)(21,24);(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,21)(17,20)(18,23)(22,25)(24,26)(27,28);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,16)(14,20)(15,24)(17,25)(18,26)(19,27)(21,23)(22,28);(13,17,18,27)(14,21,22,24)(15,20,23,28)(16,25,26,19)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7000545, #7000549, #7000554, … |
| maximal normal subgroup of | #7000545, #7000549, #7000554, … |
| maximal quotient of | #535327, #535534, #535950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38205);