Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(11,15)(14,16)(17,18,21,23)(19,20,22,24);(1,2,4,6)(3,5,8,7)(9,10,13,12)(11,14,15,16)(17,19)(18,20)(21,22)(23,24);(9,11,13,15)(10,14,12,16)(17,20)(18,22)(19,23)(21,24);(1,3)(2,5)(4,8)(6,7);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,16,15,14)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37977, #7635202, #7635310, … |
| maximal normal subgroup of | C2×#37977, #7635202, #7635310, … |
| maximal quotient of | #521334, #522071, #522683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30965);