Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(10,11)(13,14,18,21)(15,19,22,26)(16,17,20,25)(23,24,27,28);(5,6,9,7)(8,10,12,11)(15,22)(19,26)(23,27)(24,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28);(5,8,9,12)(6,10,7,11)(13,16)(14,17)(15,23)(18,20)(19,24)(21,25)(22,27)(26,28);(13,17)(14,20)(15,24)(16,21)(18,25)(19,27)(22,28)(23,26)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #214710, #214727, #214742, … |
| maximal normal subgroup of | #214710, #214727, #214742, … |
| maximal quotient of | #517708, #517710, #517711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29009);