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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26);(5,6,8,9)(7,11,12,10)(14,17)(15,21)(20,25)(22,24);(1,2)(3,4)(6,9)(10,11)(13,15,18,21)(14,19,17,23)(16,24,26,22)(20,28,25,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,26)(14,20,17,25)(15,22,21,24)(19,27,23,28);(5,7,8,12)(6,10,9,11)(13,17,18,14)(15,23,21,19)(16,25,26,20)(22,28,24,27)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=f,d^2=f,e^2=h,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 | #132176, #247007, #247035, … |
| maximal normal subgroup of | #132176, #247007, #247035, … |
| maximal quotient of | #560511, #560523, #561614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44892);