Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1199 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)(5,6,10,11)(7,8,9,12)(13,14,18,21)(15,16,22,19)(17,20,25,27)(23,24,28,26);(6,11)(7,9)(15,22)(16,19)(17,25)(20,27);(1,2)(3,4)(5,7,10,9)(6,12,11,8)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(5,8)(6,7)(9,11)(10,12)(13,16,18,19)(14,15,21,22)(17,24,25,26)(20,23,27,28);(5,9,10,7)(6,8,11,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97349, #133337, #133345, … |
| maximal normal subgroup of | #97349, #133337, #133345, … |
| maximal quotient of | #557046, #557367, #557577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39124);