Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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×H(3,𝔽4) | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1094 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)(5,6,10,11)(7,8,9,12)(14,18)(15,21)(20,25)(22,27);(6,11)(7,9)(14,18)(15,21)(16,23)(26,28);(1,2,3,4)(5,7,10,9)(6,12,11,8)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,8)(6,7)(9,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(5,9,10,7)(6,8,11,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|a^2=f,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97360, #133551, #133552, … |
| maximal normal subgroup of | #97360, #133551, #133552, … |
| maximal quotient of | #557084, #557382, #557718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39225);