Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7513941 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2749 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,8,11,7,12,6,13)(3,14,16,19,17,4,9,23)(5,10,20,28,21,27,22,26)(15,18,24,31,30,25,29,32);(1,3)(2,4)(5,15)(6,16)(7,17)(8,9)(10,25)(11,19)(12,14)(13,23)(18,27)(20,29)(21,30)(22,24)(26,32)(28,31);(1,4,7,14)(2,9)(3,11)(5,18,21,25)(6,19,8,23)(10,24)(12,16)(13,17)(15,26)(20,31,22,32)(27,29)(28,30);(1,5)(2,10)(3,15)(4,18)(6,20)(7,21)(8,22)(9,24)(11,26)(12,27)(13,28)(14,25)(16,29)(17,30)(19,31)(23,32);(1,6,7,8)(2,11,12,13)(3,16,17,9)(4,19,14,23)(5,20,21,22)(10,26,27,28)(15,29,30,24)(18,31,25,32)Polycyclic presentation:
2^9|a^2=hi,c^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,c^b=ch,d^b=df,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421140);