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 | |
1236 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)(14,18)(15,21)(20,25)(22,27);(5,6)(7,11)(8,9)(10,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28);(1,2)(3,4)(6,9)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(5,7)(6,10)(8,12)(9,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^8|b^2=f,d^2=f,e^2=f,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 | #215016, #6352349, #6352471, … |
| maximal normal subgroup of | #215016, #6352349, #6352471, … |
| maximal quotient of | #544490, #545055, #545417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33490);