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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1285 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)(17,18)(19,27)(20,31)(21,28)(22,32)(23,26)(24,29)(25,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,30,24,31,21,22)(18,25,28,27,29,32,26,20);(1,4,5,2)(3,7,8,6)(10,13)(14,15)(17,20,24,27)(18,19,29,31)(21,32,23,25)(22,28,30,26);(3,8)(6,7)(9,11,12,16)(10,14,13,15)(17,21,24,23)(18,26,29,28)(19,30,31,22)(20,25,27,32);(17,22,23,19,24,30,21,31)(18,27,28,32,29,20,26,25)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=ghi,d^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,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, 7513837);