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) = C27×C4 |
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 | |
1169 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)(5,6,8,10)(7,11,12,9)(13,14,18,21)(15,19,22,17)(16,20,26,28)(23,27,24,25);(5,7,8,12)(6,9,10,11)(15,22)(17,19)(23,24)(25,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,24)(17,21)(18,22)(20,25)(23,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(13,17,18,19)(14,15,21,22)(16,25,26,27)(20,23,28,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,e^2=fh,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 | #6668216, #6668217, #6668240, … |
| maximal normal subgroup of | #6668216, #6668217, #6668240, … |
| maximal quotient of | #544412, #545001, #545372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35631);