Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C23×#1202 (order 8192) · Inn(G) = C25 · Out(G) = C23×H(5,𝔽2) |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,24,21,22,27,29)(19,23,26,30,25,28,31,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,22)(23,28)(24,29)(30,32);(2,8)(6,9)(7,16)(14,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(19,25)(23,28)(26,31)(30,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cf,e^d=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129698, #210272, #210273, … |
| maximal normal subgroup of | #129698, #210272, #210273, … |
| maximal quotient of | #6370225, #6370226, #6370330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53242);