Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1465 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)(5,6,10,15,11,16,9,12)(7,17,21,13,18,27,20,29)(8,22,24,30,25,26,23,14)(19,28,34,35,32,36,33,31);(6,12,16,15)(7,18)(13,27,29,17)(14,26,30,22)(19,32)(20,21)(28,35,36,31)(33,34);(2,4)(5,7,9,20,11,18,10,21)(6,13,15,27,16,29,12,17)(8,19,23,33,25,32,24,34)(14,28,22,35,30,36,26,31);(5,8)(6,14)(7,19)(9,23)(10,24)(11,25)(12,26)(13,28)(15,22)(16,30)(17,31)(18,32)(20,33)(21,34)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,c^b=ci,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307286 |
| maximal normal subgroup of | #72307286 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300065);