Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,14,20,12,18,15,31)(11,21,22,37,23,16,25,33)(13,27,29,35,26,17,30,19)(24,39,38,32,28,34,40,36);(1,2)(3,7)(4,8)(5,6)(9,11,14,22,12,23,15,25)(10,16,20,33,18,21,31,37)(13,28,29,40,26,24,30,38)(17,34,19,36,27,39,35,32);(1,3,5,4)(2,7,6,8)(10,17)(11,22)(13,29)(14,15)(16,32)(18,27)(19,31)(20,35)(21,36)(23,25)(24,28)(26,30)(33,39)(34,37);(1,4,5,3)(2,7,6,8)(11,23)(16,21)(22,25)(24,28)(32,36)(33,37)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ei,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309713, #72310198, #72310206, … |
| maximal normal subgroup of | #72309713, #72310198, #72310206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405619);