Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,32,30,34,29,28)(27,35,39,36,37,33,38,40);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(26,32)(28,34)(29,31)(33,40)(35,36)(38,39);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27,30,37)(26,33,34,35)(28,36,32,40)(29,38,31,39);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,28,31,26,30,32,29,34)(27,36,39,33,37,40,38,35);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,37)(33,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=fhi,b^2=gi,c^2=hi,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308755, #72308949, #72309044 |
| maximal normal subgroup of | #72308755, #72308949, #72309044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105010);