Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2403 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,12,13)(18,22)(20,24);(2,3)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,22)(23,24);(10,16)(11,12)(18,22)(19,23);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,16,15)(17,19,21,23)(18,20,22,24);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^8|a^2=fh,d^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133514, #500075, #500423, … |
| maximal normal subgroup of | #133514, #500075, #500423, … |
| maximal quotient of | #561375, #562495, #564783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41910);