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 | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1462 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)(17,18,21,22)(19,23,27,25)(20,28,30,24)(26,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,26)(28,32)(30,31);(9,12,13,11)(10,14,15,16)(18,22)(19,25)(23,27)(24,28)(26,29)(31,32);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(17,20,21,30)(18,24,22,28)(19,26,27,32)(23,31,25,29)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gi,d^2=f,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, 6600261);