Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1534 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)(3,8)(4,7)(5,6)(17,18,23,32,22,25,21,26)(19,27,31,24,28,30,29,20);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,28)(18,24,25,20)(21,31,23,29)(26,30,32,27);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(18,25)(19,28)(26,32)(29,31);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,20,22,24)(18,19,25,28)(21,30,23,27)(26,29,32,31);(18,25)(20,24)(26,32)(27,30)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=gh,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6369174);