Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7487874 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #52004 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1852 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)(3,13)(4,10)(5,8)(6,11)(7,12)(9,15)(14,28)(16,21)(17,22)(18,25)(19,24)(20,27)(23,29)(26,30)(31,32);(2,8)(4,17)(5,6)(9,13)(10,25)(11,12)(14,29)(15,19)(18,20)(21,24)(28,32)(30,31);(1,3)(2,9)(4,14)(5,15)(6,19)(7,16)(8,21)(10,26)(11,13)(12,24)(17,29)(18,31)(20,30)(22,28)(23,25)(27,32);(1,4,5,17,7,18,6,20)(2,10,8,22,12,25,11,27)(3,14,15,29,16,31,19,30)(9,23,21,32,24,26,13,28);(2,8,12,11)(9,21,24,13)(10,22,25,27)(23,32,26,28)Polycyclic presentation:
2^9|d^2=fh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7395070);