Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | #2361 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1212 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)(8,17,23,30)(9,24)(10,14)(11,15)(12,22,19,21)(13,25)(18,27)(20,28,34,32)(26,29)(31,35,36,33);(1,2)(3,4)(5,7,10,6,11,14,16,15)(8,19,22,30,23,12,21,17)(9,18,26,13,27,24,25,29)(20,32,33,36,34,28,35,31);(2,4)(5,8,10,21,11,23,16,22)(6,12,14,30,15,19,7,17)(9,20,26,35,27,34,25,33)(13,28,24,36,29,32,18,31);(5,9)(6,13)(7,18)(8,20)(10,26)(11,27)(12,28)(14,24)(15,29)(16,25)(17,31)(19,32)(21,35)(22,33)(23,34)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281892);