Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,18,21)(15,23,25,37)(16,27,29,38)(17,30,33,22)(19,31,35,26)(20,32,36,28)(24,34,40,43)(39,42,44,41);(1,2,3,4)(6,10)(8,11)(13,15,18,25)(14,19,21,35)(16,24,29,40)(17,31,33,26)(20,34,36,43)(22,37,30,23)(27,39,38,44)(28,42,32,41);(5,6)(7,8)(9,10)(11,12)(13,16,18,29)(14,20,21,36)(15,24,25,40)(17,32,33,28)(19,34,35,43)(22,38,30,27)(23,39,37,44)(26,42,31,41);(5,7,9,12)(6,11,10,8)(15,25)(19,35)(23,37)(24,40)(26,31)(34,43)(39,44)(41,42);(5,8,9,11)(6,7,10,12)(14,21)(19,35)(20,36)(22,30)(23,37)(27,38)(34,43)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6528273);