Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| 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 | C2×#127 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1883 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)(4,5)(6,8)(11,12)(14,16)(17,18,21,24)(19,25,27,22)(20,23,28,31)(26,32,29,30);(2,5)(6,7)(9,10,13,15)(11,16,12,14)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29);(9,11,13,12)(10,14,15,16);(9,12,13,11)(10,14,15,16)(17,19,21,27)(18,22,24,25)(20,29,28,26)(23,32,31,30);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7477062);