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 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1125 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)(13,14,18,21)(15,17,23,19)(16,20,26,28)(22,25,24,27);(5,6)(7,9)(8,11)(10,12)(13,15,18,23)(14,19,21,17)(16,22,26,24)(20,27,28,25);(1,2)(3,4)(6,7)(10,11)(13,15)(14,19)(16,24)(17,21)(18,23)(20,25)(22,26)(27,28);(5,7,9,6)(8,11,12,10)(13,16)(14,20)(15,22)(17,25)(18,26)(19,27)(21,28)(23,24);(5,8,9,12)(6,10,7,11)(13,17,18,19)(14,15,21,23)(16,25,26,27)(20,22,28,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=fh,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 | #132060, #246951, #246955, … |
| maximal normal subgroup of | #132060, #246951, #246955, … |
| maximal quotient of | #560374, #560376, #561471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44995);