Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×C22⋊C4 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1628 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)(5,6,11,16,12,18,10,17)(7,19,22,34,23,33,21,13)(8,24,26,14,27,32,25,35)(9,28,29,31,30,36,20,15);(5,7)(6,13,18,34)(8,9)(10,21)(11,22)(12,23)(14,28,35,36)(15,32,31,24)(16,19,17,33)(20,25)(26,29)(27,30);(2,4)(5,8)(6,14)(7,20)(9,22)(10,25)(11,26)(12,27)(13,31)(15,34)(16,32)(17,24)(18,35)(19,36)(21,30)(23,29)(28,33);(5,9,10,20,12,30,11,29)(6,15,16,28,18,31,17,36)(7,8,21,25,23,27,22,26)(13,32,19,35,34,24,33,14)Polycyclic presentation:
2^9|a^2=h,b^2=eh,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=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272281);