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 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,13)(14,16)(18,23)(19,26)(20,21)(24,32)(25,29)(27,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,23,22,25,20,29)(19,27,28,32,26,24,30,31);(1,4,5,2)(3,7,8,6)(10,13)(14,15)(17,19,22,26)(18,24,25,27)(20,30,21,28)(23,31,29,32);(3,8)(6,7)(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,23,25,29)(19,28,26,30)(24,32,27,31);(17,18,20,29,22,25,21,23)(19,27,30,31,26,24,28,32)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=ghi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7513845);