Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 |
| prime radical | #29721 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(19,26)(25,31)(27,30)(29,32);(9,11)(10,13)(12,16)(14,15)(18,22)(19,27)(20,23)(24,28)(25,31)(26,30);(2,5)(6,8)(10,14)(11,16)(17,18,23,28,21,24,20,22)(19,25,30,32,26,31,27,29);(10,14)(13,15)(18,24)(22,28)(25,31)(29,32);(1,3,4,7)(2,6,5,8)(17,19,20,27,21,26,23,30)(18,25,22,29,24,31,28,32)Polycyclic presentation:
2^9|a^2=g,c^2=fhi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72313513, #72313671, #72314047 |
| maximal normal subgroup of | #72313513, #72313671, #72314047 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6739727);