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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,17,12,18,11,16)(7,19,21,13,22,31,28,32)(8,14,24,35,25,30,29,34)(9,26,27,15,23,36,20,33);(5,7,12,22)(6,13)(8,23,25,9)(10,21,11,28)(14,33)(15,30)(16,19)(17,31)(18,32)(20,29,27,24)(26,35)(34,36);(2,4)(5,8)(6,14)(7,20)(9,21)(10,24)(11,29)(12,25)(13,26)(15,31)(16,34)(17,35)(18,30)(19,33)(22,27)(23,28)(32,36);(5,9,10,27,12,23,11,20)(6,15,16,26,18,33,17,36)(7,8,21,24,22,25,28,29)(13,30,19,35,32,14,31,34)Polycyclic presentation:
2^9|a^2=h,b^2=efh,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=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272515);