Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353301 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #63039 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2790 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,22,18)(20,25,28,23)(21,24,31,26)(27,30,32,29);(17,20,22,28)(18,23,19,25)(21,30,31,29)(24,32,26,27);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260660);