Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,14,9,24)(7,12,19,25,18,27,16,29)(8,20,22,30,23,13,21,28)(17,32,31,35,34,26,33,36);(1,2)(3,4)(7,16,18,19)(12,25,27,29)(17,33,34,31)(26,36,32,35);(2,4)(5,7)(6,12,14,27)(8,17)(9,19)(10,16)(11,18)(13,26,20,32)(15,29,24,25)(21,31)(22,33)(23,34)(28,35,30,36);(5,8,11,23)(6,13,14,20)(7,17,18,34)(9,22,10,21)(12,26,27,32)(15,30,24,28)(16,31,19,33)(25,35,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291490);