Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,10,15,18,14)(7,17,20,11,21,28,26,13)(8,12,23,32,24,33,25,31)(19,34,22,30,35,29,36,27);(1,2)(3,4)(5,7,18,26,10,21,9,20)(6,11,16,28,15,13,14,17)(8,22,25,19,24,36,23,35)(12,29,32,27,33,34,31,30);(2,4)(5,8,10,24)(6,12)(7,19,21,35)(9,25,18,23)(11,27)(13,30)(14,32)(15,33)(16,31)(17,34)(20,36,26,22)(28,29);(5,7,10,21)(6,13,15,11)(8,19,24,35)(9,26,18,20)(12,30,33,27)(14,17,16,28)(22,23,36,25)(29,32,34,31)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=eh,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 295661);