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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,20,26)(15,21,29,23)(16,30,32,41)(17,35,28,37)(18,25,19,24)(22,34,40,33)(27,38,42,44)(31,36,43,39);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15,20,29)(14,21,26,23)(16,31,32,43)(17,18,28,19)(22,38,40,44)(24,35,25,37)(27,34,42,33)(30,36,41,39);(5,8,9,6)(7,11,12,10)(13,16,19,34,20,32,18,33)(14,22,25,41,26,40,24,30)(15,27,28,43,29,42,17,31)(21,36,35,44,23,39,37,38);(7,12)(10,11)(13,17,19,15,20,28,18,29)(14,23,25,37,26,21,24,35)(16,31,34,27,32,43,33,42)(22,39,41,38,40,36,30,44);(1,3,4,2)(16,32)(22,40)(27,42)(30,41)(31,43)(33,34)(36,39)(38,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=h,d^2=gi,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511721);