Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C43 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C2×#342 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,16,27)(8,19,20,12)(9,22)(10,14)(11,17)(13,23)(18,32,30,28)(21,26,34,33)(24,29)(25,36)(31,35);(5,7,9,18,10,16,23,30)(6,11,13,25,14,24,22,35)(8,17,21,31,20,29,34,36)(12,15,26,28,19,27,33,32);(1,2,3,4)(7,16)(11,24)(15,27)(17,29)(18,30)(25,35)(28,32)(31,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,f^2=i,b^a=bd,d^a=dh,f^a=fi,h^a=hi,c^b=ci,e^b=ehi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312051, #72312068, #72312079, … |
| maximal normal subgroup of | #72312051, #72312068, #72312079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53559);