Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1696 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,20)(19,26,30,24)(21,31,23,25)(27,29,32,28);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,20,26)(21,29,23,28)(25,32,31,27);(10,14)(13,15)(17,20,22,18)(19,27,30,32)(21,31,23,25)(24,28,26,29);(2,7)(3,8)(11,16)(13,15)(17,21,22,23)(18,25,20,31)(19,28,30,29)(24,27,26,32);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,25,20,31)(19,29,30,28)(24,32,26,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419106);