Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1298 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)(5,6,11,16,12,17,10,27)(7,18,21,31,22,30,20,13)(8,23,25,33,26,15,9,14)(19,28,24,34,36,29,35,32);(5,7)(6,13,17,31)(8,24)(9,19)(10,20)(11,21)(12,22)(14,32,33,34)(15,29,23,28)(16,18,27,30)(25,36)(26,35);(2,4)(5,8,10,9,12,26,11,25)(6,14,16,23,17,33,27,15)(7,19,21,24,22,36,20,35)(13,28,30,32,31,29,18,34);(5,9,10,26,12,25,11,8)(6,15,16,14,17,23,27,33)(7,19,20,35,22,36,21,24)(13,29,18,32,31,28,30,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275622);