Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(6,10)(7,8)(13,14,18,24,17,23,19,25)(15,26,29,37,28,20,30,21)(16,31,34,22,33,41,35,40)(27,39,43,42,32,38,44,36);(5,6,9,10)(7,11,8,12)(13,15,19,30,17,28,18,29)(14,20,25,37,23,26,24,21)(16,27,34,43,33,32,35,44)(22,36,41,39,40,42,31,38);(2,4)(5,7)(6,11)(8,9)(10,12)(13,15,17,28)(14,21,23,37)(16,32,33,27)(18,30,19,29)(20,25,26,24)(22,39,40,38)(31,42,41,36)(34,43,35,44);(5,8,9,7)(6,12,10,11)(13,16,17,33)(14,22,23,40)(15,27,28,32)(18,35,19,34)(20,36,26,42)(21,38,37,39)(24,31,25,41)(29,44,30,43)Polycyclic presentation:
2^9|a^2=efi,b^2=eg,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372021);