Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188770 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
877 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)(7,16,19,11)(8,20,22,13)(9,12)(10,14)(15,23)(17,25)(18,26,32,24)(21,28,33,27)(29,31)(30,34)(35,36);(1,2)(3,4)(5,7)(6,11)(8,17)(9,18)(10,19)(12,24)(13,25)(14,16)(15,26)(20,29)(21,30)(22,31)(23,32)(27,34)(28,35)(33,36);(2,4)(6,12,14,15)(11,24,16,26)(13,27,20,28)(25,34,29,35);(5,8,9,21,10,22,23,33)(6,13,12,27,14,20,15,28)(7,17,18,30,19,31,32,36)(11,25,24,34,16,29,26,35)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268421);