Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2826 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1338 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)(3,7)(4,17)(5,10)(6,8)(9,20)(11,12)(13,24)(14,28)(15,23)(16,21)(18,25)(19,26)(22,27)(29,32)(30,31);(1,3,12,13,5,16,6,15)(2,7,8,23,10,21,11,24)(4,14,20,27,19,29,18,30)(9,22,17,28,25,31,26,32);(2,8)(3,13)(4,18)(6,12)(7,21)(10,11)(14,29)(15,16)(17,26)(19,20)(22,32)(28,31);(1,4)(2,9)(3,14)(5,19)(6,18)(7,22)(8,17)(10,25)(11,26)(12,20)(13,27)(15,30)(16,29)(21,31)(23,28)(24,32);(2,10)(7,21)(8,11)(9,25)(17,26)(22,31)(23,24)(28,32)Polycyclic presentation:
2^9|b^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,d^c=dfh,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538652);