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 |
| prime radical | C2×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1225 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)(13,14,18,22)(15,19,23,28)(16,20,26,25)(17,21,27,24);(6,9)(10,11)(15,23)(17,27)(19,28)(21,24);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,21)(17,25)(18,23)(20,27)(22,28)(24,26);(5,7,8,12)(6,10,9,11)(13,16)(14,20)(15,24)(17,19)(18,26)(21,23)(22,25)(27,28);(13,17)(14,21)(15,25)(16,19)(18,27)(20,23)(22,24)(26,28)Polycyclic presentation:
2^8|a^2=fh,c^2=g,d^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 | #482888, #6353241, #6353251, … |
| maximal normal subgroup of | #482888, #6353241, #6353251, … |
| maximal quotient of | #544515, #545077, #545500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34395);