Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,23,28)(19,29,30,32)(20,25,24,31)(21,27,22,26);(9,10)(11,14)(12,13)(15,16)(17,19,21,31,23,30,22,25)(18,24,26,32,28,20,27,29);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20)(18,25)(19,26)(21,29)(22,32)(23,24)(27,30)(28,31);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(19,30)(20,24)(25,31)(29,32);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,28,27)(19,31,30,25)(20,29,24,32)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403105);