Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,9)(8,22,26,31)(10,16)(11,15)(12,17)(13,19)(14,25,30,24)(18,32,35,29)(20,27)(21,28)(23,34,36,33);(1,2)(3,4)(5,7,11,20,12,21,10,19)(6,13,16,28,17,27,15,9)(8,18,25,34,26,35,24,33)(14,29,31,36,30,32,22,23);(5,8,10,24,12,26,11,25)(6,14,16,31,17,30,15,22)(7,18,19,33,21,35,20,34)(9,23,13,29,28,36,27,32);(2,4)(5,9,10,13,12,28,11,27)(6,7,16,19,17,21,15,20)(8,23,24,29,26,36,25,32)(14,18,31,33,30,35,22,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,f^c=fi,g^c=gi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314246, #72314281 |
| maximal normal subgroup of | #72314246, #72314281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264191);