Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,18,11,17,10,16)(7,19,9,13,22,15,21,30)(8,23,26,34,27,14,28,33)(20,31,25,29,35,32,24,36);(1,2)(3,4)(5,7,12,21,11,22,10,9)(6,13,18,19,17,30,16,15)(8,24,28,20,27,25,26,35)(14,31,34,29,23,32,33,36);(2,4)(5,8)(6,14)(7,20)(9,25)(10,28)(11,27)(12,26)(13,29)(15,32)(16,34)(17,23)(18,33)(19,31)(21,24)(22,35)(30,36);(5,9,10,22,11,21,12,7)(6,15,16,30,17,19,18,13)(8,25,28,35,27,24,26,20)(14,32,34,36,23,31,33,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=fh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285128);