Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1070 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)(5,6)(7,12)(8,10)(9,11)(13,14,18,16)(15,19,21,23)(17,20,25,24)(22,26,27,28);(5,6,10,8)(7,11,9,12)(15,21)(17,25)(19,23)(20,24);(1,2,3,4)(5,7,10,9)(6,11,8,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27);(5,8,10,6)(7,11,9,12)(13,16,18,14)(15,19,21,23)(17,24,25,20)(22,26,27,28);(5,9,10,7)(6,11,8,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=fh,e^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 | C2×#42178, #7640698, #7640703, … |
| maximal normal subgroup of | C2×#42178, #7640698, #7640703, … |
| maximal quotient of | #544535, #545091, #545531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35166);