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 | |
1425 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)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32);(9,12,13,11)(10,14,15,16)(17,19,22,20)(18,23,25,27)(21,28,29,26)(24,32,31,30);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(17,20,22,19)(18,23,25,27)(21,26,29,28)(24,32,31,30)Polycyclic presentation:
2^9|a^2=g,b^2=g,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, 6600257);