Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,18,23)(15,24,28,20)(16,29,19,21)(17,31,33,22)(25,37,26,35)(27,36,43,41)(30,38,34,40)(32,39,42,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,23,24)(16,26,19,25)(17,32,33,42)(21,35,29,37)(22,39,31,44)(27,34,43,30)(36,38,41,40);(1,2,3,4)(15,25)(16,19)(20,35)(21,29)(24,37)(26,28)(27,32)(30,34)(36,39)(38,40)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,23,29)(15,26,28,25)(17,30,33,34)(20,35,24,37)(22,38,31,40)(27,42,43,32)(36,39,41,44);(6,9)(10,12)(13,17,18,33)(14,22,23,31)(15,27,28,43)(16,30,19,34)(20,36,24,41)(21,38,29,40)(25,32,26,42)(35,39,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590743);