Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2563 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
960 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)(7,16,20,12)(8,21)(9,11)(10,15)(13,22)(14,24)(17,25,19,28)(18,32,34,35)(23,26)(27,33,30,31)(29,36);(1,2)(3,4)(6,11,15,14)(7,17,20,19)(13,26,29,21)(18,31,34,33);(2,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,25)(13,27)(14,28)(15,16)(17,24)(21,35)(22,33)(23,34)(26,32)(29,30)(31,36);(5,8,9,22,10,23,24,36)(6,13,14,21,15,29,11,26)(7,18,19,33,20,34,17,31)(12,27,28,35,16,30,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272442);