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 | C22×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1392 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)(5,6,10,14,11,16,9,28,12,17,8,24,26,35,27,15)(7,18,22,33,23,30,21,31,19,34,20,36,29,32,25,13);(1,2,3,4)(5,7)(6,13,17,31)(8,25)(9,22)(10,21)(11,29)(12,19)(14,30,24,32)(15,34,28,18)(16,36,35,33)(20,27)(23,26);(6,14)(8,9)(10,27)(11,26)(13,30)(15,16)(17,24)(18,33)(20,21)(22,25)(23,29)(28,35)(31,32)(34,36);(7,19)(13,31)(18,34)(20,22)(21,25)(23,29)(30,32)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,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 | #72231505, #72231541, #72231551, … |
| maximal normal subgroup of | #72231505, #72231541, #72231551, … |
| maximal quotient of | #72250246, C2×#494560, #72309569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401756);