Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #495677 (order 512) · Inn(G) = C22⋊C4 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22⋊C4 |
| commutator | C22 | C2×C32 |
| Frattini, cosocle | C2×C32 | C22 |
| socle | C2 | C22⋊C32 |
| prime radical | C23 | C32 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,9,4,10,14,27,5,11,15,28,17,30,36,49,6,12,16,29,18,31,37,50,20,32,38,51,39,52,56,62,7,8,13,23,19,24,33,43,21,25,34,44,40,46,53,58,22,26,35,45,41,47,54,59,42,48,55,60,57,61,63,64);(2,8)(3,13)(10,24)(11,25)(12,26)(14,33)(15,34)(16,35)(30,46)(31,47)(32,48)(36,53)(37,54)(38,55)(52,61)(56,63)Polycyclic presentation:
2^8|a^2=d,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bc,c^a=ch,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #94512, #94513, #94577, … |
| maximal normal subgroup of | #94512, #94513, #94577, … |
| maximal quotient of | #94510, #94575, #94604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 501);