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×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,9,13)(7,15)(8,19,20,33)(10,24,23,14)(11,26)(12,29,30,22)(16,31)(17,28)(18,35)(21,32)(25,34)(27,36);(2,3)(7,16,18,25)(8,20)(11,27,28,32)(12,30)(15,31,35,34)(17,21,26,36)(19,33)(22,29);(5,7)(6,11)(8,21)(9,18)(10,25)(12,31)(13,28)(14,32)(15,33)(16,23)(17,29)(19,35)(20,36)(22,26)(24,27)(30,34);(1,3,4,2)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=e,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 | #72286580, #72286646, D4×#2180, … |
| maximal normal subgroup of | #72286580, #72286646, D4×#2180, … |
| maximal quotient of | #72287175, #72287433, #72309488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405528);