Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1780 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)(7,17,20,13,21,27,19,28)(8,14,24,29,22,30,23,25)(18,31,34,26,32,35,33,36);(2,4)(6,12,16,15)(8,22)(13,17,28,27)(14,29,30,25)(18,32)(23,24)(26,35,36,31)(33,34);(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,17)(14,26)(15,27)(16,28)(22,32)(23,33)(24,34)(25,31)(29,35)(30,36);(5,8,10,24,11,22,9,23)(6,14,15,29,16,30,12,25)(7,18,20,34,21,32,19,33)(13,26,27,35,28,36,17,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,f^c=fi116 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, 261515);