Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #23854 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,24)(19,23)(25,33)(26,34)(27,39)(28,35)(29,36)(30,37)(31,38)(32,40)(41,43)(42,44);(6,9)(7,11)(13,15)(14,20)(16,29)(17,26)(18,27)(19,28)(21,36)(22,34)(23,35)(24,39)(25,30)(31,41)(32,42)(33,37)(38,43)(40,44);(2,4)(13,16,19,31)(14,21,23,38)(15,25,28,42)(17,30,18,32)(20,33,35,44)(22,37,24,40)(26,41,27,29)(34,43,39,36);(6,9)(10,12)(13,17,19,18)(14,22,23,24)(15,26,28,27)(16,30,31,32)(20,34,35,39)(21,37,38,40)(25,41,42,29)(33,43,44,36);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(17,18)(20,34)(21,38)(25,29)(26,28)(30,32)(33,43)(35,39)(36,44)(41,42)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580190);