Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355925 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,4)(3,10,11,16)(5,9,6,15)(8,13,12,14)(17,18)(19,28)(20,24)(21,26)(22,27)(23,31)(25,30)(29,32);(1,3,5,13,7,11,6,14)(2,8,15,16,4,12,9,10)(17,19,22,20)(18,24,27,28)(21,30,23,29)(25,26,32,31);(3,11)(8,12)(10,16)(13,14)(17,20,22,19)(18,25,27,32)(21,30,23,29)(24,26,28,31);(1,4,7,2)(3,12,11,8)(5,9,6,15)(10,14,16,13)(17,21,22,23)(18,26,27,31)(19,29,20,30)(24,25,28,32);(1,5,7,6)(2,9,4,15)(3,13,11,14)(8,10,12,16)(18,27)(24,28)(25,32)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=i,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263160);