Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | #23850 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,27)(25,40)(26,35)(28,41)(29,44)(30,37)(31,39)(32,38)(34,42)(36,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(2,4)(13,15,18,27,19,28,17,26)(14,20,22,24,23,35,33,41)(16,29,30,43,31,25,32,42)(21,36,38,44,39,34,37,40);(5,7,8,11)(6,10,9,12)(13,16,19,31)(14,21,23,39)(15,25,28,29)(17,32,18,30)(20,34,35,36)(22,38,33,37)(24,40,41,44)(26,43,27,42);(2,4)(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,30,31,32)(20,24,35,41)(21,37,39,38)(25,42,29,43)(34,44,36,40)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590849);