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 | |
1074 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)(13,14,18,15)(16,19,24,21)(17,22,25,20)(23,27,28,26);(5,6)(7,9)(8,11)(10,12)(14,15)(17,25)(19,21)(23,28);(1,2,3,4)(6,7)(10,11)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(5,7,9,6)(8,11,12,10)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(5,8)(6,10)(7,11)(9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=f,c^2=fg,d^2=fh,e^2=f,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 | #132260, #245925, #245926, … |
| maximal normal subgroup of | #132260, #245925, #245926, … |
| maximal quotient of | #560455, #560486, #561552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43682);