Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C2×C4 | C82 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1273 |
| prime radical | C23 | #322 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,8,12,15,6,10,13,16,3,7,11,14)(18,22)(19,25)(24,29)(26,31);(2,7,10,8)(9,14,16,15)(17,18,19,23,20,24,26,30,21,22,25,28,27,29,31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=ch,e^a=ehi,c^b=ci152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#92909, #72210674 |
| maximal normal subgroup of | C2×#92909, #72210674 |
| maximal quotient of | #72210244, C2×#92909 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105);