Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,10,16,9,12)(7,13,19,25,20,29,17,28)(8,21,24,30,23,26,22,14)(18,32,33,27,34,35,31,36);(1,2)(3,4)(6,12,16,15)(7,17,20,19)(14,26,30,21)(18,31,34,33);(2,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,17)(12,25)(14,27)(15,28)(16,29)(21,32)(22,33)(23,34)(24,31)(26,35)(30,36);(5,8,9,22,10,23,11,24)(6,14,15,21,16,30,12,26)(7,18,19,33,20,34,17,31)(13,27,28,32,29,36,25,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272247);