Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,24)(16,28)(17,22)(18,33)(19,23)(20,25)(21,29)(26,35)(27,36)(30,38)(31,39)(32,40)(34,37)(41,43)(42,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(15,25)(16,29)(17,18)(20,24)(21,28)(22,33)(27,34)(31,32)(36,37)(39,40)(41,44)(42,43);(13,15,19,27)(14,20,23,37)(16,30,32,26)(17,25,18,34)(21,38,39,35)(22,36,33,24)(28,44,40,41)(29,43,31,42);(2,4)(13,16,19,32)(14,21,23,39)(15,26,27,30)(17,29,18,31)(20,35,37,38)(22,40,33,28)(24,41,36,44)(25,42,34,43);(6,10)(9,11)(13,17,19,18)(14,22,23,33)(15,25,27,34)(16,31,32,29)(20,36,37,24)(21,28,39,40)(26,43,30,42)(35,41,38,44)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421983);