Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1293 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)(7,11)(8,9)(10,12)(13,14,18,19)(15,21,24,20)(16,25,27,17)(22,28,26,23);(6,9)(10,11)(14,19)(17,25)(20,21)(23,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,20,19,21)(16,26,27,22)(17,23,25,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,17)(15,22)(18,27)(19,25)(20,28)(21,23)(24,26);(13,17)(14,16)(15,23)(18,25)(19,27)(20,26)(21,22)(24,28)Polycyclic presentation:
2^8|a^2=fh,c^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131343, #484163, #484249, … |
| maximal normal subgroup of | #131343, #484163, #484249, … |
| maximal quotient of | #544503, #545066, #545487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34392);