Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,23,25)(19,26,21,22)(20,28,30,24)(27,31,29,32);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,18,23,21,26,25)(20,29,31,28,30,27,32,24);(1,3,4,8)(2,5,6,7)(17,20,23,30)(18,24,25,28)(19,27,21,29)(22,31,26,32);(2,6)(5,7)(11,15)(13,16)(18,25)(19,21)(20,30)(31,32);(17,21,22,25,23,19,26,18)(20,29,31,28,30,27,32,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313310, #72313393, #72313550, … |
| maximal normal subgroup of | #72313310, #72313393, #72313550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497608);