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 | C2×#555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1027 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)(13,14,18,17)(15,21,23,19)(16,24,26,20)(22,27,25,28);(5,6,9,7)(8,11,12,10)(14,17)(19,21)(20,24)(27,28);(1,2)(3,4)(6,7)(10,11)(13,15,18,23)(14,19,17,21)(16,25,26,22)(20,28,24,27);(5,7,9,6)(8,11,12,10)(13,16,18,26)(14,20,17,24)(15,22,23,25)(19,27,21,28);(5,8,9,12)(6,10,7,11)(13,17,18,14)(15,21,23,19)(16,24,26,20)(22,28,25,27)Polycyclic presentation:
2^8|a^2=f,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 | #132188, #248824, #248825, … |
| maximal normal subgroup of | #132188, #248824, #248825, … |
| maximal quotient of | #544426, #545013, #545364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35953);