Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,13,15)(11,14,12,16)(17,18,22,23)(19,24,29,32)(20,26,21,25)(27,30,28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,29)(23,32)(25,27)(26,28);(9,12,13,11)(10,14,15,16)(17,20,22,21)(18,25,23,26)(19,27,29,28)(24,31,32,30);(1,3,4,7)(2,6,5,8)(17,18,22,23)(19,24,29,32)(20,25,21,26)(27,31,28,30);(2,5)(6,8)(17,21,22,20)(18,26,23,25)(19,28,29,27)(24,30,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6600267);