Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,16,20)(11,21,25,26)(12,14,19,15)(13,18)(17,24,33,23)(22,37)(27,29)(28,40)(30,35)(31,36)(32,38)(34,39);(1,2)(3,7)(4,8)(5,6)(9,11,16,25)(10,17,20,33)(12,26,19,21)(13,28,31,38)(14,24,15,23)(18,34,36,37)(22,30,39,29)(27,40,35,32);(1,3,5,4)(2,7,6,8)(9,12,15,10,16,19,14,20)(11,17,24,26,25,33,23,21)(13,27,29,18,31,35,30,36)(22,32,28,34,39,40,38,37);(1,4,5,3)(2,7,6,8)(9,13)(10,18)(11,22)(12,27)(14,30)(15,29)(16,31)(17,32)(19,35)(20,36)(21,37)(23,38)(24,28)(25,39)(26,34)(33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=efg,d^2=ef,e^2=i,f^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353194);