Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1638 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)(5,6,10,17,12,18,11,16)(7,19,22,13,23,31,25,32)(8,24,26,14,21,34,30,35)(9,27,20,15,29,33,28,36);(5,7,12,23)(6,13)(8,9,21,29)(10,22,11,25)(14,33)(15,24)(16,19)(17,31)(18,32)(20,30,28,26)(27,35)(34,36);(2,4)(5,8)(6,14)(7,20)(9,25)(10,26)(11,30)(12,21)(13,15)(16,24)(17,34)(18,35)(19,27)(22,29)(23,28)(31,33)(32,36);(5,9,10,20,12,29,11,28)(6,15,16,27,18,36,17,33)(7,21,22,30,23,8,25,26)(13,24,19,35,32,34,31,14)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272280);