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 |
| prime radical | C2×#548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1154 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)(13,14,18,19)(15,23,25,20)(16,26,27,21)(17,22,28,24);(5,6)(7,11)(8,9)(10,12)(14,19)(17,28)(20,23)(21,26);(1,2,3,4)(6,9)(10,11)(13,15,18,25)(14,20,19,23)(16,22,27,24)(17,21,28,26);(5,6)(7,10)(8,9)(11,12)(13,16,18,27)(14,21,19,26)(15,24,25,22)(17,23,28,20);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,21)(16,23)(18,28)(19,24)(20,27)(25,26)Polycyclic presentation:
2^8|a^2=f,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 | #482904, #6353141, #6353151, … |
| maximal normal subgroup of | #482904, #6353141, #6353151, … |
| maximal quotient of | #544518, #545076, #545507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33577);