Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,11,17,27,16)(7,18,21,33,22,34,26,31)(8,13,9,24,23,14,25,32)(12,19,28,20,30,35,29,36);(1,2,3,4);(5,7,11,22)(6,12,17,30)(8,20,23,36)(9,19,25,35)(10,26,27,21)(13,31,14,33)(15,29,16,28)(18,24,34,32);(5,8,10,9,11,23,27,25)(6,13,15,24,17,14,16,32)(7,19,21,20,22,35,26,36)(12,18,28,33,30,34,29,31)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285383, #72286595, #72286631, … |
| maximal normal subgroup of | #72285383, #72286595, #72286631, … |
| maximal quotient of | #72285410, #72285431, #72285438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403040);