Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,18)(9,22)(10,25)(11,14)(13,16)(15,27)(17,33)(19,32)(20,36)(21,28)(23,31)(24,29)(26,30)(34,35);(1,2)(3,4)(5,7,11,9)(6,12,14,22)(8,19,21,35)(10,24,27,17)(13,30,31,36)(15,33,25,29)(16,26,23,20)(18,32,28,34);(1,3,4,2)(5,8)(6,13)(7,16)(9,23)(10,26)(11,21)(12,28)(14,31)(15,34)(17,35)(18,22)(19,24)(20,27)(25,32)(29,36)(30,33);(6,14)(12,22)(13,31)(15,25)(18,28)(29,33)(30,36)(32,34)Polycyclic presentation:
2^9|b^2=i,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406305);