Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 131072 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
990 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)(15,20)(16,24)(17,22)(18,19)(21,27)(23,26)(25,28);(5,6,9,7)(8,11,12,10)(14,19)(17,20)(21,24)(26,28);(1,2)(3,4)(6,7)(10,11)(13,15,18,22)(14,20,19,17)(16,25,27,23)(21,26,24,28);(5,7,9,6)(8,11,12,10)(13,16,18,27)(14,21,19,24)(15,23,22,25)(17,26,20,28);(5,8,9,12)(6,10,7,11)(13,17,18,20)(14,22,19,15)(16,26,27,28)(21,25,24,23)Polycyclic presentation:
2^8|b^2=h,c^2=f,d^2=fh,e^2=fh,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 | #132196, #248826, #248837, … |
| maximal normal subgroup of | #132196, #248826, #248837, … |
| maximal quotient of | #544427, #545012, #545365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35952);