Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2362 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,27,29,24)(20,25,31,32)(21,26,28,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,24,23,27)(20,30,31,26)(21,25,28,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,31)(18,25,23,32)(19,28,29,21)(24,26,27,30);(17,21)(18,26)(19,20)(22,28)(23,30)(24,32)(25,27)(29,31);(19,29)(20,31)(24,27)(25,32)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348215);