Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1019 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)(7,11)(10,12)(13,14,18,22)(15,20,16,19)(17,25,26,21)(23,27,24,28);(6,9)(10,12)(13,15,18,16)(14,19,22,20)(17,24,26,23)(21,28,25,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,20,22,19)(17,24,26,23)(21,28,25,27);(5,6)(7,10)(8,9)(11,12)(13,17,18,26)(14,21,22,25)(15,23,16,24)(19,27,20,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98056, #261204, #261208, … |
| maximal normal subgroup of | #98056, #261204, #261208, … |
| maximal quotient of | #553763, #553767, #554020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29730);