Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,18,23)(15,26,28,21)(16,30,32,40)(17,35,29,37)(19,25,20,24)(22,34,39,33)(27,42,43,38)(31,41,44,36);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,28)(14,21,23,26)(16,31,32,44)(17,19,29,20)(22,38,39,42)(24,37,25,35)(27,34,43,33)(30,36,40,41);(5,8,9,6)(7,11,12,10)(13,16,20,34,18,32,19,33)(14,22,25,40,23,39,24,30)(15,27,29,44,28,43,17,31)(21,36,37,42,26,41,35,38);(7,12)(10,11)(13,17,20,15,18,29,19,28)(14,21,25,37,23,26,24,35)(16,31,34,27,32,44,33,43)(22,36,40,42,39,41,30,38);(1,3,4,2)(14,23)(16,32)(21,26)(24,25)(27,43)(31,44)(33,34)(35,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,d^2=gi,e^2=fi,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511720);