Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1138 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)(3,4)(5,6,10,17,12,18,11,27)(7,19,23,30,20,34,24,28)(8,16,25,14,26,32,9,15)(13,22,31,21,29,36,33,35);(2,4)(5,7)(6,13,18,29)(8,22)(9,21)(10,24)(11,23)(12,20)(14,28,15,30)(16,19,32,34)(17,33,27,31)(25,35)(26,36);(6,14)(8,25)(9,26)(10,11)(13,28)(15,18)(16,17)(19,33)(21,36)(22,35)(23,24)(27,32)(29,30)(31,34);(7,20)(13,29)(19,34)(21,35)(22,36)(23,24)(28,30)(31,33)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231475, #72231532, #72231551, … |
| maximal normal subgroup of | #72231475, #72231532, #72231551, … |
| maximal quotient of | #72256837, C2×#494340, #72309709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401536);