Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| 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×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,7)(9,10,13,18,14,20,28,19)(11,15,22,31,23,33,39,32)(12,24,26,17,27,16,25,34)(21,35,37,30,38,29,36,40);(1,2,4,5)(3,7,8,6)(9,11,14,23)(10,15,20,33)(12,21,27,38)(13,22,28,39)(16,29,24,35)(17,30,34,40)(18,31,19,32)(25,36,26,37);(10,16)(12,25)(13,28)(15,29)(17,18)(19,34)(20,24)(21,36)(22,39)(26,27)(30,31)(32,40)(33,35)(37,38);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=g,b^2=ei,d^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309480, #72309494, #72309504, … |
| maximal normal subgroup of | #72309480, #72309494, #72309504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404819);