Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| 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 | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,3)(5,6)(7,8)(9,10)(11,23)(12,29)(13,32)(14,21)(15,20)(16,22)(17,26)(18,31)(19,34)(24,37)(25,38)(27,36)(28,35)(30,39)(33,40);(2,3)(9,11,14,26,16,28,15,27)(10,17,20,23,22,36,21,35)(12,24,30,34,25,13,31,33)(18,37,38,40,39,32,29,19);(9,12,16,25)(10,18,22,39)(11,24,28,13)(14,31,15,30)(17,32,36,37)(19,23,40,35)(20,29,21,38)(26,33,27,34);(1,3,4,2)(6,8)(9,13,16,24)(10,19,22,40)(11,25,28,12)(14,34,15,33)(17,29,36,38)(18,23,39,35)(20,32,21,37)(26,30,27,31)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=i,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337643);