Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,15,10,16,24,12,11,17,9,14,20,35,26,30)(7,18,21,32,22,34,27,33,23,29,19,31,28,36,25,13);(1,2)(3,4)(6,12,17,30)(7,19,23,21)(8,24)(9,26)(10,20)(13,31)(14,16,15,35)(22,27,28,25)(32,33)(34,36);(2,4)(5,7,20,22,11,23,10,28)(6,13,16,31,17,33,35,32)(8,25,26,21,9,27,24,19)(12,29,14,34,30,18,15,36)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250226, #72250238, #72250252, … |
| maximal normal subgroup of | #72250226, #72250238, #72250252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56595);