Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1577 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)(11,15)(12,13)(14,16)(17,18,24,28)(19,22,27,23)(20,30,32,26)(21,25,31,29);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,27)(29,32)(30,31);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,23,21,24,32,22,31)(18,25,27,30,28,29,19,26);(1,4,5,3)(2,6,8,7)(9,11,12,16)(10,14,13,15)(17,21,23,32,24,31,22,20)(18,26,27,25,28,30,19,29);(2,8)(6,7)(17,22,24,23)(18,27,28,19)(20,31,32,21)(25,30,29,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503450);