Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(6,9)(10,11)(13,14)(15,24)(16,21)(17,29)(18,33)(19,23)(20,27)(22,31)(25,40)(26,42)(28,39)(30,36)(32,41)(34,38)(35,43)(37,44);(5,6,8,9)(7,11,12,10)(15,25)(20,35)(24,40)(26,34)(27,43)(30,44)(36,37)(38,42);(13,15,19,25)(14,20,23,35)(16,26,18,34)(17,30,32,44)(21,36,33,37)(22,38,39,42)(24,41,40,29)(27,28,43,31);(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,23,33)(15,26,25,34)(17,28,32,31)(20,36,35,37)(22,29,39,41)(24,42,40,38)(27,30,43,44)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404813);