Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,10,15,9,25)(7,16,20,27,17,30,19,12)(8,21,24,28,22,35,23,13)(18,32,34,36,31,26,33,29);(1,2,3,4)(5,7,9,19,10,17,11,20)(6,12,14,16,15,27,25,30)(8,18,23,33,22,31,24,34)(13,26,21,29,28,32,35,36);(2,4)(7,17)(8,22)(13,28)(19,20)(21,35)(23,24)(26,32)(29,36);(5,8,9,23,10,22,11,24)(6,13,14,21,15,28,25,35)(7,18,19,33,17,31,20,34)(12,26,16,29,27,32,30,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273642);