Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1322 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)(5,6,11,17,12,18,10,28,13,19,9,27,8,16,26,14)(7,20,24,15,21,31,23,32,25,35,22,33,30,36,29,34);(1,2,3,4)(6,14,19,28)(8,12)(9,26)(10,11)(15,31,33,36)(16,17,18,27)(20,34,35,32)(21,30)(22,29)(23,24);(5,7)(6,15)(8,21)(9,23)(10,22)(11,29)(12,30)(13,25)(14,31)(16,32)(17,20)(18,34)(19,33)(24,26)(27,35)(28,36);(5,8,13,12)(6,16,19,18)(7,21,25,30)(9,10,11,26)(14,27,28,17)(15,32,33,34)(20,31,35,36)(22,29,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231517, #72231535, #72231551, … |
| maximal normal subgroup of | #72231517, #72231535, #72231551, … |
| maximal quotient of | #72249948, C2×#494562, #72309586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401758);