Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #34624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,27,22,19,21,25)(20,24,30,32,28,26,29,31);(9,10)(11,15)(12,13)(14,16)(17,19,22,18)(20,26,28,24)(21,25,23,27)(29,31,30,32);(2,6)(7,8)(10,13)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(20,28)(24,26)(29,30)(31,32);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=g,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376914);