Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#382 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 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)(7,15,17,28)(8,19,21,12)(9,13)(10,14)(11,18,24,32)(16,29)(20,34,35,25)(22,26)(23,31)(27,30)(33,36);(5,7)(6,11,14,24)(8,16)(9,18)(10,17)(12,23,19,33)(13,15,26,28)(20,31)(21,30)(22,32)(25,27,34,29)(35,36);(2,4)(5,8,9,20,10,21,22,35)(6,12,13,25,14,19,26,34)(7,16,18,31,17,30,32,36)(11,23,15,27,24,33,28,29);(7,17)(11,24)(15,28)(16,30)(18,32)(23,33)(27,29)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=di,g^b=gi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250462, #72250486, #72250496, … |
| maximal normal subgroup of | #72250462, #72250486, #72250496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260693);