Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,31)(20,24,21,25)(27,30,29,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(10,14)(13,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(1,3,4,7)(2,6,5,8);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72282648, #72283330, #72283935, … |
| maximal normal subgroup of | #72282648, #72283330, #72283935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339683);