Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62729 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,22,24,32)(13,18)(14,17)(16,23,30,25)(19,33,34,40)(20,28)(21,29)(26,31)(27,35,38,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,27)(17,28)(18,29)(22,33)(23,35)(24,34)(25,37)(26,36)(30,38)(31,39)(32,40);(9,12,13,25,14,24,26,23)(10,16,18,32,17,30,31,22)(11,19,21,37,20,34,36,35)(15,27,29,40,28,38,39,33);(3,8)(5,7)(10,17)(11,20)(16,30)(18,31)(19,34)(21,36)(22,32)(35,37);(2,6)(5,7)(11,20)(15,28)(19,34)(21,36)(27,38)(29,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=h,c^2=gh,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300566);