Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483962 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C22×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,16,20,19)(11,21,24,23)(12,26)(13,28)(14,27)(17,33)(18,30,31,32)(22,37,25,38)(35,36)(39,40);(2,5)(6,8)(9,10,13,19,14,18,28,30,15,20,12,16,27,31,26,32)(11,17,21,34,25,35,37,40,24,36,23,29,22,33,38,39);(1,3,4,7)(2,6,5,8)(9,11,12,23,14,25,26,38,15,24,13,21,27,22,28,37)(10,17,16,29,18,35,32,39,20,36,19,34,31,33,30,40);(10,18,20,31)(11,22,24,25)(12,13)(16,30,19,32)(21,37,23,38)(26,28)(29,34)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dhi,g^b=gi,d^c=dh,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265106, #72265116, #72265409, … |
| maximal normal subgroup of | #72265106, #72265116, #72265409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403258);