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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,28)(21,26)(23,25)(24,31)(27,29)(30,32);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(18,23)(19,26)(21,29)(22,30)(25,31)(27,28);(9,12,13,16)(10,14,11,15)(18,23)(22,25)(24,32)(30,31);(18,23)(19,27)(21,29)(22,31)(25,30)(26,28)Polycyclic presentation:
2^9|b^2=gi,c^2=fg,d^2=fi,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, 6598266);