Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14)(15,22)(16,19)(17,25)(18,21)(20,26)(23,27)(24,28);(6,9)(10,12)(13,15,18,16)(14,19,21,22)(17,23,26,24)(20,27,25,28);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,19,21,22)(17,24,26,23)(20,27,25,28);(5,7,8,11)(6,10,9,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,27)(25,26);(1,3,4,2)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #97009, #209156, #209157, … |
| maximal normal subgroup of | #97009, #209156, #209157, … |
| maximal quotient of | #513878, #513880, #513962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27516);