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×#3197 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2055 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)(15,20)(16,26)(17,22)(18,32)(19,23)(21,24)(25,34)(27,36)(28,39)(29,33)(30,37)(31,42)(35,38)(40,44)(41,43);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,23,32)(16,27,29,24)(17,25,30,31)(21,26,36,33)(22,34,37,42)(28,40,41,38)(35,39,44,43);(2,4)(13,16,19,29)(14,21,23,36)(15,24,18,27)(17,28,30,41)(20,33,32,26)(22,35,37,44)(25,38,31,40)(34,43,42,39);(6,9)(10,12)(13,17)(14,22)(15,25)(16,28)(18,31)(19,30)(20,34)(21,35)(23,37)(24,38)(26,39)(27,40)(29,41)(32,42)(33,43)(36,44);(2,4)(5,7,8,11)(6,10,9,12)(17,30)(22,37)(25,31)(28,41)(34,42)(35,44)(38,40)(39,43)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6580324);