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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,18,21)(15,19,24,33)(16,26,28,36)(17,29,31,22)(20,30,35,27)(23,37,38,44)(25,41,40,34)(32,42,43,39);(1,2,3,4)(6,10)(8,11)(13,15,18,24)(14,19,21,33)(16,23,28,38)(17,25,31,40)(20,32,35,43)(22,34,29,41)(26,37,36,44)(27,39,30,42);(5,6)(7,8)(9,10)(11,12)(13,16,18,28)(14,20,21,35)(15,23,24,38)(17,30,31,27)(19,32,33,43)(22,36,29,26)(25,42,40,39)(34,44,41,37);(5,7,9,12)(6,11,10,8)(15,24)(19,33)(23,38)(25,40)(32,43)(34,41)(37,44)(39,42);(5,8,9,11)(6,7,10,12)(14,21)(19,33)(20,35)(22,29)(26,36)(32,43)(34,41)(37,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6525072);