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:
(1,2,3,4)(5,6,11,15)(7,16,21,30)(8,9,23,12)(10,25,27,14)(13,19,32,18)(17,29,31,26)(20,33,35,34)(22,28,36,24);(5,7)(6,13,15,32)(8,16,23,30)(9,18)(10,26)(11,21)(12,19)(14,33,25,34)(17,22,31,36)(20,28)(24,35)(27,29);(5,8,9,15,11,23,12,6)(7,17,18,33,21,31,19,34)(10,14,24,22,27,25,28,36)(13,29,16,20,32,26,30,35)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250234, #72250248, #72250252, … |
| maximal normal subgroup of | #72250234, #72250248, #72250252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56163);