Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22,26,30,27,23,25,29)(24,32,34,28,35,36,33,31);(1,2,3,4)(5,7,8,14,9,13,15,19)(6,10,11,17,12,16,18,20)(21,23,27,22)(24,31,35,28)(25,29,26,30)(32,34,36,33);(21,24,27,35)(22,28,23,31)(25,34,26,33)(29,36,30,32);(7,13)(10,16)(14,19)(17,20)(22,23)(28,31)(29,30)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100809);