Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58990 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3007 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)(3,17)(4,22)(5,23)(6,26)(7,15)(8,14)(9,16)(10,19)(11,24)(12,25)(13,27)(18,29)(20,31)(21,30)(28,32);(1,3,9,21)(2,10,16,31)(4,23,13,29)(5,6,18,11)(7,20,8,19)(12,26,32,24)(14,30,15,17)(22,28,27,25);(1,4)(2,11)(3,12)(5,17)(6,16)(7,24)(8,26)(9,13)(10,28)(14,22)(15,27)(18,30)(19,23)(20,29)(21,32)(25,31);(1,5,9,18)(2,12,16,32)(3,6,21,11)(4,17,13,30)(7,25,8,28)(10,24,31,26)(14,23,15,29)(19,27,20,22);(1,6)(2,13)(3,18)(4,16)(5,21)(7,22)(8,27)(9,11)(10,29)(12,30)(14,24)(15,26)(17,32)(19,25)(20,28)(23,31)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,g^c=gi,h^c=hi,g^d=gi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421339);