Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,28,36)(16,22,31,40)(17,23,18,21)(25,34,29,38)(26,35,27,33)(30,37,32,39)(41,43,42,44);(1,2)(3,4)(6,9)(7,11)(13,15,19,28)(14,20,24,36)(16,29,31,25)(17,26,18,27)(21,33,23,35)(22,38,40,34)(30,42,32,41)(37,44,39,43);(2,4)(14,21)(17,18)(20,33)(22,37)(23,24)(26,27)(30,32)(34,43)(35,36)(38,44)(39,40)(41,42);(6,9)(10,12)(13,16)(14,22)(15,25)(17,32)(18,30)(19,31)(20,34)(21,37)(23,39)(24,40)(26,42)(27,41)(28,29)(33,43)(35,44)(36,38);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,21)(15,26,28,27)(16,30,31,32)(20,35,36,33)(22,37,40,39)(25,41,29,42)(34,43,38,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^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,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580481);