Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14,17,21)(15,23,25,19)(16,27,29,43)(18,31,28,22)(20,34,35,30)(24,38,40,42)(26,33,39,37)(32,41,36,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,19,21,23)(16,24,29,40)(18,26,28,39)(20,32,35,36)(22,33,31,37)(27,42,43,38)(30,41,34,44);(2,4)(6,9)(10,11)(13,16,28,34,17,29,18,30)(14,20,31,43,21,35,22,27)(15,24,39,44,25,40,26,41)(19,32,37,38,23,36,33,42);(5,7,8,12)(6,10,9,11)(14,21)(15,25)(16,29)(22,31)(26,39)(30,34)(32,36)(38,42);(2,4)(15,25)(19,23)(24,40)(26,39)(32,36)(33,37)(38,42)(41,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592130);