Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792416 |
| 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 | |
1826 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)(4,17)(5,21)(6,10)(7,11)(8,16)(9,20)(13,23)(14,28)(15,25)(18,31)(19,24)(22,27)(26,29)(30,32);(3,13)(4,18)(5,6)(8,12)(9,24)(10,21)(15,16)(17,31)(19,20)(23,25)(26,28)(27,30);(1,3)(2,8)(4,14)(5,13)(6,16)(7,15)(9,22)(10,25)(11,23)(12,21)(17,32)(18,30)(19,27)(20,29)(24,28)(26,31);(1,2)(3,8)(4,9)(5,21)(6,10)(7,11)(12,13)(14,22)(15,23)(16,25)(17,20)(18,24)(19,31)(26,27)(28,30)(29,32);(1,4,6,18,7,20,5,19)(2,9,10,24,11,17,21,31)(3,14,16,30,15,29,13,27)(8,22,25,28,23,32,12,26)Polycyclic presentation:
2^9|e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699612);