Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,20)(16,27)(17,32)(18,34)(19,23)(21,31)(22,29)(24,37)(25,40)(26,36)(28,35)(30,44)(33,38)(39,41)(42,43);(1,2)(3,4)(6,9)(10,11)(13,15,19,26)(14,20,23,36)(16,28,30,33)(17,29,25,18)(21,37,39,43)(22,40,34,32)(24,41,42,31)(27,35,44,38);(13,16,29,41,19,30,18,31)(14,21,34,44,23,39,22,27)(15,24,17,33,26,42,25,28)(20,35,40,43,36,38,32,37);(2,4)(13,17,18,15,19,25,29,26)(14,20,22,32,23,36,34,40)(16,28,31,42,30,33,41,24)(21,38,27,43,39,35,44,37);(5,7,8,12)(6,10,9,11)(16,30)(21,39)(24,42)(27,44)(28,33)(31,41)(35,38)(37,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=g,e^2=f,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,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470143);