Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16,19,33)(8,13)(9,21)(10,25)(11,15)(12,18,29,36)(14,24)(17,31,26,34)(20,30)(22,32)(23,28,35,27);(2,3)(7,17,19,26)(9,22)(12,27,29,28)(14,30)(16,31,33,34)(18,23,36,35)(20,24)(21,32);(5,7)(6,12)(8,17)(9,23)(10,26)(11,19)(13,28)(14,31)(15,29)(16,32)(18,20)(21,33)(22,35)(24,36)(25,27)(30,34);(1,3,4,2)(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,22,24)(12,28,29,27)(14,21,30,32)(16,31,33,34)(18,35,36,23)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=ei,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250241, #72250863, #72250867, … |
| maximal normal subgroup of | #72250241, #72250863, #72250867, … |
| maximal quotient of | #72287601, #72287699, #72287754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405529);