Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1184 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)(13,14,19,22)(15,21,17,20)(16,25,26,18)(23,28,27,24);(5,6)(7,11)(8,9)(10,12)(13,15,19,17)(14,20,22,21)(16,23,26,27)(18,28,25,24);(1,2)(3,4)(6,9)(10,11)(13,16)(14,18)(15,23)(17,27)(19,26)(20,28)(21,24)(22,25);(5,6)(7,10)(8,9)(11,12)(13,17,19,15)(14,21,22,20)(16,23,26,27)(18,28,25,24);(5,7)(6,10)(8,12)(9,11)(13,18)(14,16)(15,24)(17,28)(19,25)(20,27)(21,23)(22,26)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #214777, #6352373, #6352374, … |
| maximal normal subgroup of | #214777, #6352373, #6352374, … |
| maximal quotient of | #544442, #545026, #545387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33479);