Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#476 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19)(8,23,27,30)(9,15)(10,17)(11,28)(12,18)(13,21)(14,26,32,20)(16,22)(24,33,25,35)(29,31,36,34);(1,2)(3,4)(5,7,10,21,12,22,9,11)(6,13,15,19,18,28,17,16)(8,20,25,36,27,26,24,29)(14,30,31,35,32,23,34,33);(2,4)(5,8,10,25,12,27,9,24)(6,14,15,31,18,32,17,34)(7,20,21,36,22,26,11,29)(13,30,19,35,28,23,16,33);(5,9,12,10)(6,15,18,17)(7,11,22,21)(8,24,27,25)(13,19,28,16)(14,31,32,34)(20,29,26,36)(23,33,30,35)Polycyclic presentation:
2^9|a^2=eh,b^2=e,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307101 |
| maximal normal subgroup of | #72307101 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406403);