Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(15,23)(16,28)(17,25)(19,24)(20,29)(21,31)(26,40)(27,32)(30,35)(33,41)(34,39)(36,42);(2,4)(5,7,9,8)(6,10,12,11)(13,14,18,22)(15,24,27,39)(16,29,30,42)(17,31,25,21)(19,32,34,23)(20,35,36,28)(26,41,40,33)(37,38,43,44);(13,15,25,32,18,27,17,23)(14,19,31,39,22,34,21,24)(16,26,35,43,30,40,28,37)(20,33,42,44,36,41,29,38);(1,2,3,4)(13,16,17,28,18,30,25,35)(14,20,21,29,22,36,31,42)(15,26,23,37,27,40,32,43)(19,33,24,38,34,41,39,44);(5,8,9,7)(6,11,12,10)(16,30)(20,36)(26,40)(28,35)(29,42)(33,41)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gi,d^2=h,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^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521372);