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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1257 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)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,28)(22,33)(24,32)(25,31)(27,35)(29,34)(30,39)(36,38)(37,40);(2,3)(9,11,13,23,15,25,14,24)(10,16,18,21,20,32,19,31)(12,22,27,37,29,39,28,38)(17,33,34,36,35,30,26,40);(9,12,15,29)(10,17,20,35)(11,22,25,39)(13,28,14,27)(16,30,32,33)(18,26,19,34)(21,36,31,40)(23,38,24,37);(1,3,4,2)(6,8)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,30)(18,31)(19,21)(20,32)(26,40)(27,38)(28,37)(29,39)(33,35)(34,36)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=i,d^2=ef,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, 337642);