Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1828 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)(5,6,10,17,12,18,11,16)(7,19,22,13,23,31,28,32)(8,14,9,27,26,15,25,34)(20,30,24,29,21,33,36,35);(5,7,12,23)(6,13)(8,24,26,36)(9,21,25,20)(10,22,11,28)(14,33)(15,30)(16,19)(17,31)(18,32)(27,35)(29,34);(1,2,3,4)(5,8,11,25,12,26,10,9)(6,14,16,34,18,15,17,27)(7,20,22,24,23,21,28,36)(13,29,31,33,32,35,19,30);(5,9,10,26,12,25,11,8)(6,15,16,27,18,14,17,34)(7,21,22,36,23,20,28,24)(13,30,19,35,32,33,31,29)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276365);