Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | D4×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,19,21,35)(9,23)(10,15)(12,20,28,33)(13,17)(14,18)(16,32,22,29)(24,31,36,26)(25,27)(30,34);(1,2)(3,4)(5,7,10,18)(6,11,15,14)(8,16,21,22)(9,17,25,34)(12,26,28,31)(13,27,30,23)(19,32,35,29)(20,24,33,36);(5,8)(6,12)(7,16)(9,24)(10,21)(11,26)(13,29)(14,31)(15,28)(17,33)(18,22)(19,27)(20,34)(23,35)(25,36)(30,32);(2,4)Polycyclic presentation:
2^9|a^2=g,b^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250863, #72250872, #72250885, … |
| maximal normal subgroup of | #72250863, #72250872, #72250885, … |
| maximal quotient of | #72287582, #72287682, #72287740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404588);