Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(11,20,24,23)(12,25,27,26)(16,21,30,34)(17,31,33,32)(22,37)(29,39)(35,38)(36,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,21)(12,17)(13,18)(14,28)(15,19)(16,23)(20,30)(22,36)(24,34)(25,31)(26,32)(27,33)(29,38)(35,39)(37,40);(1,4,5,6)(2,3,7,8)(9,11)(10,16)(12,22)(13,23)(14,20)(15,24)(17,29)(18,34)(19,30)(21,28)(25,38)(26,35)(27,37)(31,40)(32,36)(33,39);(2,7)(3,8)(9,12)(10,17)(11,22)(13,25)(14,26)(15,27)(16,29)(18,31)(19,33)(20,35)(21,36)(23,38)(24,37)(28,32)(30,39)(34,40)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290499);