Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136735 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,15,10,17,27,16)(7,11,20,31,22,28,21,34)(8,23,25,13,26,12,24,14)(18,30,35,32,36,29,19,33);(1,2,3,4)(5,7)(6,11)(8,18)(9,20)(10,22)(12,29)(13,32)(14,33)(15,31)(16,34)(17,28)(19,24)(21,27)(23,30)(25,35)(26,36);(6,12)(7,18)(8,24)(9,27)(11,28)(13,15)(14,16)(17,23)(19,20)(21,35)(22,36)(25,26)(29,33)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250148, #72250154, #72250157, … |
| maximal normal subgroup of | #72250148, #72250154, #72250157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55304);