Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7339796 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #48088 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23872 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1399 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)(3,11,14,26)(4,9)(5,10)(6,8)(7,13,20,19)(12,27,29,24)(15,21,30,31)(16,22)(17,23)(18,25)(28,32);(1,3)(2,7)(4,15)(5,19)(6,14)(8,20)(9,24)(10,26)(11,23)(12,28)(13,17)(16,30)(18,29)(21,32)(22,27)(25,31);(2,8)(4,16)(5,17)(7,20)(12,29)(13,19)(24,27)(25,32);(1,4,17,28,6,16,5,18)(2,9,23,32,8,22,10,25)(3,12,13,30,14,29,19,15)(7,21,11,27,20,31,26,24);(1,3)(2,7)(4,12)(5,19)(6,14)(8,20)(9,21)(10,26)(11,23)(13,17)(15,18)(16,29)(22,31)(24,25)(27,32)(28,30)Polycyclic presentation:
2^9|a^2=gi,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7246993);