Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #29814 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1387 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)(3,6)(4,7)(5,8)(9,10,13,15)(11,14,12,16);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,14)(13,15)(17,18,22,25)(19,26,20,28)(21,30,31,24)(23,29,32,27);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,14,15,16)(17,19,22,20)(18,23,25,32)(21,29,31,27)(24,28,30,26);(9,12,13,11)(10,14,15,16)(19,20)(23,32)(26,28)(27,29);(1,4,5,3)(2,7,8,6)(17,20,22,19)(18,23,25,32)(21,27,31,29)(24,28,30,26)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=fgi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6635533);