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×#449 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1177 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)(21,22,25,27,26,30,24,29)(23,31,32,28,34,35,33,36);(1,2,3,4)(5,7,8,14,9,13,15,19)(6,10,11,17,12,16,18,20)(22,27)(24,25)(28,31)(29,30)(32,33)(35,36);(21,23)(22,28)(24,32)(25,33)(26,34)(27,31)(29,35)(30,36);(7,13)(10,16)(14,19)(17,20)(21,24,26,25)(22,27,30,29)(23,32,34,33)(28,31,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^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, 100810);