Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792410 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #38904 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,9)(5,20)(6,10)(7,11)(8,16)(13,22)(14,26)(15,25)(17,23)(18,30)(19,24)(21,29)(27,31)(28,32);(3,13)(4,17)(5,6)(8,12)(9,23)(10,20)(15,16)(18,19)(22,25)(24,30)(26,32)(27,29);(1,3)(2,8)(4,14)(5,13)(6,16)(7,15)(9,21)(10,25)(11,22)(12,20)(17,29)(18,27)(19,28)(23,32)(24,31)(26,30);(1,2)(3,8)(4,9)(5,20)(6,10)(7,11)(12,13)(14,21)(15,22)(16,25)(17,23)(18,30)(19,24)(26,27)(28,31)(29,32);(1,4,6,17,7,19,5,18)(2,9,10,23,11,24,20,30)(3,14,16,29,15,28,13,27)(8,21,25,32,22,31,12,26)Polycyclic presentation:
2^9|e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699606);