Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,28)(19,27,23,24)(20,29,31,32)(21,30,25,26);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,28,27)(20,30,31,26)(21,32,25,29);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,20,19,26,22,31,23,30)(18,25,24,32,28,21,27,29);(2,6)(5,7)(10,14)(13,15)(17,21)(18,26)(19,29)(20,27)(22,25)(23,32)(24,31)(28,30);(3,8)(5,7)(17,19,22,23)(18,27,28,24)(20,26,31,30)(21,32,25,29)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815250);