Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2058 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,12,15,13,17,10,29)(7,9,16,11,19,28,30,27)(8,20,25,33,26,21,23,14)(18,35,34,24,32,22,31,36);(1,2,3,4)(5,7,9,15,10,16,27,17,13,19,28,29,12,30,11,6)(8,21,22,32,25,33,36,34,26,20,35,18,23,14,24,31);(5,8,10,23,13,26,12,25)(6,14,15,20,17,33,29,21)(7,18,16,34,19,32,30,31)(9,24,27,35,28,36,11,22)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249869, #72249871, #72249879, … |
| maximal normal subgroup of | #72249869, #72249871, #72249879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56445);